Natural Lacunae Method and Schatten–Von Neumann Classes of the Convergence Exponent

Abstract

Our first aim is to clarify the results obtained by Lidskii devoted to the decomposition on the root vector system of the non-selfadjoint operator. We use a technique of the entire function theory and introduce a so-called Schatten–von Neumann class of the convergence exponent. Considering strictly accretive operators satisfying special conditions formulated in terms of the norm, we construct a sequence of contours of the power type that contrasts the results by Lidskii, where a sequence of contours of the exponential type was used.

1. Introduction

Generally, the concept originates from the well-known fact that the eigenvectors of the compact selfadjoint operator form a basis in the closure of its range. The question of what happens in the case when the operator is non-selfadjoint is rather complicated and deserves to be considered as a separate part of the spectral theory. Basically, the aim of the mentioned part of the spectral theory is propositions on the convergence of the root vector series in one or another sense to an element belonging to the closure of the operator range. Here, we should note when we say a sense, we mean Bari, Riesz, Abel (Abel–Lidskii) senses of the series convergence [1,2]. A reasonable question that appears is about minimal conditions that guarantee the desired result. For instance, in the mentioned papers, the authors considered a domain of the parabolic type containing the spectrum of the operator. In the paper [1], non-salfadjoint operators with the special condition imposed on the numerical range of values are considered. The main advantage of this result is a weak condition imposed upon the numerical range of values comparatively with the sectorial condition (see the definition of the sectorial operator). Thus, the convergence in the Abel–Lidskii sense was established for an operator class wider than the class of sectorial operators. Here, we make a comparison between results devoted to operators with the discrete spectra and operators with the compact resolvent, for they can be easily reformulated from one to another realm.

The central idea of this paper is to formulate sufficient conditions of the Abel–Lidskii basis property of the root functions system for a sectorial non-selfadjoint operator of the special type. Considering such an operator class, we slightly strengthen the condition regarding the semi-angle of the sector, but we weaken a great deal of conditions regarding the involved parameters. Moreover, the central aim generates some prerequisites to consider technical peculiarities such as a newly constructed sequence of contours of the power type that is in contrast to the Lidskii results [3], where a sequence of the contours of the exponential type was considered. Thus, we clarify the results [3] devoted to the decomposition on the root vector system of the non-selfadjoint operator. We use a technique of the entire function theory and introduce a so-called Schatten–von Neumann class of the convergence exponent. Considering strictly accretive operators satisfying special conditions formulated in terms of the norm, using a sequence of contours of the power type, we invent a peculiar method regarding how to calculate a contour integral involved in the problem in its general statement. Finally, we produce applications to differential equations in the abstract Hilbert space. In particular, the existence and uniqueness theorems for evolution equations with the right-hand side—a differential operator with a fractional derivative in final terms—are covered by the invented abstract method. In this regard, such operators as the Riemann–Liouville fractional differential operator, Kipriyanov operator, Riesz potential, and difference operator are involved. Note that analysis of the required conditions imposed upon the right-hand side of the evolution equations that are in the scope leads us to the relevance of the central idea of the paper. Here, we should note a well-known fact [4,5] that a particular interest appears in the case when a senior term of the operator at least is not selfadjoint, for in the contrary case, there are plenty of results devoted to the topic wherein the following papers are well-known [5,6,7,8,9]. The fact is that most of them deal with a decomposition of the operator on a sum, where the senior term must be either a selfadjoint or normal operator. In other cases, the methods of the papers [4,10] become relevant and allow us to study spectral properties of operators whether we have the above-mentioned representation or not. We should remark that the results of the papers [1,8], applicable to the study of non-selfadjoint operators, are based on the sufficiently strong assumption regarding the numerical range of values of the operator. At the same time, the methods [4] can be used in the natural way if we deal with more abstract constructions formulated in terms of the semigroup theory [11]. The central challenge of the latter paper is how to create a model representing a composition of fractional differential operators in terms of the semigroup theory. We should note that motivation arises in connection with the fact that a second-order differential operator can be represented as some kind of a transform of the infinitesimal generator of a shift semigroup. Here, we should stress that the eigenvalue problem for the operator was previously studied by methods of theory of functions [12]. Thus, having been inspired by the novelty of the idea, we generalize a differential operator with a fractional integro-differential composition in the final terms to some transform of the corresponding infinitesimal generator of the shift semigroup. By virtue of the methods obtained in the paper [4], we managed to study spectral properties of the infinitesimal generator transform and obtain an outstanding result—asymptotic equivalence between the real component of the resolvent and the resolvent of the real component of the operator. The relevance is based on the fact that the asymptotic formula for the operator’s real component can be established in most cases due to the well-known asymptotic relations for the regular differential operators as well as for the singular ones [13]. It is remarkable that the results establishing the spectral properties of non-selfadjoint operators allow us to implement a novel approach regarding the problem of the basis property of root vectors. In its own turn, the application of results connected with the basis property covers many problems in the framework of the theory of evolution equations. The abstract approach to the Cauchy problem for the fractional evolution equation was previously implemented in the papers [14,15]. At the same time, the main advantage of this paper is the obtained formula for the solution of the evolution equation with the relatively wide conditions imposed upon the right-hand side, where the derivative at the left-hand side is supposed to be of the fractional order. This problem appeals to many ones that lie in the framework of the theory of differential equations: for instance, in the paper [16], the solution of the evolution equation can be obtained in an analytical way if we impose the conditions upon the right-hand side. We can also produce a number of papers dealing with differential equations which can be studied by this paper’s abstract methods [17,18,19,20,21]. The latter information gives us an opportunity to claim that the offered approach is undoubtedly novel and relevant.

2. Preliminaries

Let $C,C_i,i\in {\mathbb{N}}_0$ be real constants. We assume that a value of C is positive and can be different in various formulas, but values of $C_i$ are certain. Denote by $\mathrm{int}M,\mathrm{Fr}M$ the interior and the set of boundary points of the set M, respectively. Everywhere further, if the contrary is not stated, we consider linear densely defined operators acting on a separable complex Hilbert space $\mathfrak{H}$. Denote by $\mathcal{B}\left(\mathfrak{H}\right)$ the set of linear bounded operators on $\mathfrak{H}.$ Denote by $\stackrel{\sim }{L}$ the closure of an operator $L.$ We establish the following agreement on using symbols ${\stackrel{\sim }{L}}^i:={(\stackrel{\sim }{L})}^i,$ where i is an arbitrary symbol. Denote by $\mathrm{D}\left(L\right),\mathrm{R}\left(L\right),\mathrm{and}\mathrm{N}\left(L\right)$ the domain of definition, the range, and the kernel or null space of an operator L, respectively. The deficiency (codimension) of $\mathrm{R}\left(L\right)\mathrm{and}$ dimension of $\mathrm{N}\left(L\right)$ are denoted by $\mathrm{def}L,\mathrm{nul}L$, respectively. Assume that L is a closed operator acting on $\mathfrak{H};\mathrm{N}\left(L\right)=0,$ let us define a Hilbert space ${\mathfrak{H}}_L:=\left\{f,g\in \mathrm{D}\left(L\right),{(f,g)}_{{\mathfrak{H}}_L}={(Lf,Lg)}_{\mathfrak{H}}\right\}.$ Consider a pair of complex Hilbert spaces $\mathfrak{H},{\mathfrak{H}}_{+};$ the notation ${\mathfrak{H}}_{+}\subset \subset \mathfrak{H}$ means that ${\mathfrak{H}}_{+}$ is dense in $\mathfrak{H}$ as a set of elements and we have a bounded embedding provided by the inequality

$${\parallel f\parallel }_{\mathfrak{H}}\le C_0{\parallel f\parallel }_{{\mathfrak{H}}_{+}},C_0>0,f\in {\mathfrak{H}}_{+},$$

Moreover, any bounded set with respect to the norm ${\mathfrak{H}}_{+}$ is compact with respect to the norm $\mathfrak{H}.$ Let L be a closed operator; for any closable operator S such that $\stackrel{\sim }{S}=L,$ its domain $\mathrm{D}\left(S\right)$ will be called a core of $L.$ Denote by ${\mathrm{D}}_0\left(L\right)$ a core of a closeable operator $L.$ Let $\mathrm{P}\left(L\right)$ be the resolvent set of an operator L and $R_L\left(\zeta \right),\zeta \in \mathrm{P}\left(L\right),where[R_L:=R_L\left(0\right)]$ denotes the resolvent of an operator $L.$ Denote by ${\lambda }_i\left(L\right),i\in \mathbb{N}$ the eigenvalues of an operator $L.$ Suppose L is a compact operator and $N:={\left(L^{\ast }L\right)}^{1/2},r\left(N\right):=\dim \mathrm{R}\left(N\right);$ then, the eigenvalues of the operator N are called the singular numbers (s-numbers) of the operator L and are denoted by $s_i\left(L\right),i=1,2,\dots ,r\left(N\right).$ If $r\left(N\right)<\infty ,$ then we put by definition $s_i=0,i=r\left(N\right)+1,2,\dots$. According to the terminology of the monograph [2], the dimension of the root vectors subspace corresponding to a certain eigenvalue ${\lambda }_k$ is called the algebraic multiplicity of the eigenvalue ${\lambda }_k.$ Let $\nu \left(L\right)$ denotes the sum of all algebraic multiplicities of an operator $L.$ Denote by $n\left(r\right)$ a function equal to a number of the elements of the sequence ${\left\{a_n\right\}}_1^{\infty },|a_n|\uparrow \infty$ within the circle $\left|z\right|<r.$ Let A be a compact operator, denote by $n_A\left(r\right)$counting function a function $n\left(r\right)$ corresponding to the sequence ${\left\{s_i^{-1}\left(A\right)\right\}}_1^{\infty }.$ Let ${\mathfrak{S}}_p\left(\mathfrak{H}\right),0<p<\infty$ be a Schatten–von Neumann class and ${\mathfrak{S}}_{\infty }\left(\mathfrak{H}\right)$ be the set of compact operators. Suppose L is an operator with a compact resolvent and $s_n\left(R_L\right)\le C{n}^{-\mu },n\in \mathbb{N},0\le \mu <\infty ;$ then, we denote by $\mu \left(L\right)$ order of the operator L in accordance with the definition given in the paper [5]. Denote by $\mathfrak{Re}L:=\left(L+L^{\ast }\right)/2,\mathfrak{Im}L:=\left(L-L^{\ast }\right)/2i$ the real and imaginary components of an operator L, respectively. In accordance with the terminology of the monograph [22], the set $\mathsf{\Theta }\left(L\right):=\{z\in \mathbb{C}:z={(Lf,f)}_{\mathfrak{H}},f\in \mathrm{D}\left(L\right),\parallel f{\parallel }_{\mathfrak{H}}=1\}$ is called the numerical range of an operator $L.$ An operator L is called sectorial if its numerical range belongs to a closed sector ${\mathfrak{L}}_{\iota }\left(\theta \right):=\{\zeta :|arg(\zeta -\iota )|\le \theta <\pi /2\},$ where $\iota$ is the vertex and $\theta$ is the semi-angle of the sector ${\mathfrak{L}}_{\iota }\left(\theta \right).$ If we want to stress the correspondence between $\iota$ and $\theta ,$ then we will write ${\theta }_{\iota }.$ An operator L is called bounded from below if the following relation holds $\mathrm{Re}{(Lf,f)}_{\mathfrak{H}}\ge {\gamma }_L{\parallel f\parallel }_{\mathfrak{H}}^2,f\in \mathrm{D}\left(L\right),{\gamma }_L\in \mathbb{R},$ where ${\gamma }_L$ is called a lower bound of $L.$ An operator L is called accretive if ${\gamma }_L=0.$ An operator L is called strictly accretive if ${\gamma }_L>0.$ An operator L is called m-accretive if the next relation holds ${(A+\zeta )}^{-1}\in \mathcal{B}\left(\mathfrak{H}\right),\parallel {(A+\zeta )}^{-1}\parallel \le {\left(\mathrm{Re}\zeta \right)}^{-1},\mathrm{Re}\zeta >0.$ An operator L is called m-sectorial if L is sectorial and $L+\beta$ is m-accretive for some constant $\beta .$ An operator L is called symmetric if one is densely defined and the following equality holds: ${(Lf,g)}_{\mathfrak{H}}={(f,Lg)}_{\mathfrak{H}},f,g\in \mathrm{D}\left(L\right).$

Consider a sesquilinear form $t[·,·]$ (see [22]) defined on a linear manifold of the Hilbert space $\mathfrak{H}.$ Denote by $t[·]$ the quadratic form corresponding to the sesquilinear form $t[·,·].$ Let $\mathfrak{h}=(t+t^{\ast })/2,\mathfrak{k}=(t-t^{\ast })/2i$ be a real and imaginary component of the form t, respectively, where $t^{\ast }[u,v]=t\overline{[v,u]},\mathrm{D}\left(t^{\ast }\right)=\mathrm{D}\left(t\right).$ According to these definitions, we have $\mathfrak{h}[·]=\mathrm{Re}t[·],\mathfrak{k}[·]=\mathrm{Im}t[·].$ Denote by $\stackrel{\sim }{t}$ the closure of a form $t.$ The range of a quadratic form $t\left[f\right],f\in {\mathrm{D}\left(t\right),\parallel f\parallel }_{\mathfrak{H}}=1$ is called the range of the sesquilinear form t and is denoted by $\mathsf{\Theta }\left(t\right).$ A form t is called sectorial if its range belongs to a sector having a vertex $\iota$ situated at the real axis and a semi-angle $0\le {\theta }_{\iota }<\pi /2.$ Suppose t is a closed sectorial form; then, a linear manifold ${\mathrm{D}}_0\left(t\right)\subset \mathrm{D}\left(t\right)$ is called the core of t if the restriction of t to ${\mathrm{D}}_0\left(t\right)$ has the closure t (see [22] p. 166). Due to Theorem 2.7 [22] p. 323, there exist unique m-sectorial operators $T_t,T_{\mathfrak{h}}$ associated with the closed sectorial forms $t,\mathfrak{h}$, respectively. The operator $T_{\mathfrak{h}}$ is called a real part of the operator $T_t$ and is denoted by $Re{T}_t.$ Suppose L is a sectorial densely defined operator and $t[u,v]:={(Lu,v)}_{\mathfrak{H}},\mathrm{D}\left(t\right)=\mathrm{D}\left(L\right);$ then, due to Theorem 1.27 [22] p. 318, the corresponding form t is closable, due to Theorem 2.7 [22] p. 323, there exists a unique m-sectorial operator $T_{\stackrel{\sim }{t}}$ associated with the form $\stackrel{\sim }{t}.$ In accordance with the definition [22] p. 325, the operator $T_{\stackrel{\sim }{t}}$ is called a Friedrichs extension of the operator $L.$ Everywhere further, unless otherwise stated, we use notations of the papers [2,22,23,24,25].

2.1. Some Properties of Non-Selfadjoint Operators

In this section, we explore a special operator class for which a number of spectral theory theorems can be applied. As an application of the obtained abstract results, we study a basis property of the root vectors of the operator in terms of the order of the operator real part. By virtue of such an approach, we express a convergence exponent of $\mathit{s}$-numbers through the order of the operator real part. The theorem given below (see [11]) gives us a description of spectral properties of some class of non-selfadjoint operators.

Theorem 1.

Assume that L is a non-selfadjoint operator acting in $\mathfrak{H};$ then, the following conditions hold

$\left(\mathrm{H}1\right)$ There exists a Hilbert space ${\mathfrak{H}}_{+}\subset \subset \mathfrak{H}$ and a linear manifold $\mathfrak{M}$ that is dense in ${\mathfrak{H}}_{+}.$ The operator L is defined on $\mathfrak{M}.$

$\left(\mathrm{H}2\right)\left|{(Lf,g)}_{\mathfrak{H}}\right|\le C_1{\parallel f\parallel }_{{\mathfrak{H}}_{+}}{\parallel g\parallel }_{{\mathfrak{H}}_{+}},\mathrm{Re}{(Lf,f)}_{\mathfrak{H}}\ge C_2{\parallel f\parallel }_{{\mathfrak{H}}_{+}}^2,f,g\in \mathfrak{M},C_1,C_2>0.$

Let W be a restriction of the operator L on the set $\mathfrak{M}.$ Then, the following propositions are true.

$\left(\mathbf{A}\right)$ We have the following classification

$$R_{\stackrel{\sim }{W}}\in {\mathfrak{S}}_p,p=\left\{\begin{array}{c}\hfill l,l>2/\mu ,\mu \le 1,\\ \hfill 1,\mu >1\end{array}\right.,$$

here and further, μ is the order of $H:=Re\stackrel{\sim }{W}.$ Moreover, under the assumptions ${\lambda }_n\left(R_H\right)\ge C{n}^{-\mu },n\in \mathbb{N},$ we have the following implication

$$\left\{R_{\stackrel{\sim }{W}}\in {\mathfrak{S}}_p,1\le p<\infty \right\}\Rightarrow \mu p>1.$$

$\left(\mathbf{B}\right)$ The following relation holds

$$\sum _{i=1}^n{\left|{\lambda }_i(R_{\stackrel{\sim }{W}})\right|}^p\le C\sum _{i=1}^n{\lambda }_i^p\left(R_H\right),1\le p<\infty ,(n=1,2,\dots \nu (R_{\stackrel{\sim }{W}})),$$

moreover, if $\nu (R_{\stackrel{\sim }{W}})=\infty$ and $\mu \ne 0,$ then the following asymptotic formula holds

$$|{\lambda }_i(R_{\stackrel{\sim }{W}})|=o\left(i^{-\tau }\right),i\to \infty ,0<\tau <\mu .$$

$\left(\mathbf{C}\right)$ Assume that $\theta <\pi \mu /2,$ where θ is the semi-angle of the sector ${\mathfrak{L}}_0\left(\theta \right)\supset \mathsf{\Theta }(\stackrel{\sim }{W}).$ Then the system of root vectors of $R_{\stackrel{\sim }{W}}$ is complete in $\mathfrak{H}.$

Throughout the paper, we formulate results in terms of the restriction W on the set $\mathfrak{M}$ of the operator L satisfying the Theorem 1 conditions. Here, we should note that we are motivated by plenty of applications considered in the papers [4,10,11]. The main achievement of the latter paper is a model representing a composition of fractional differential operators in terms of the semigroup theory. Thus, the operator W having the the spectral properties described in the Theorem 1 is a generalized form of the concrete operators studied in recent years. We also use the short-hand notations $A:=R_{\stackrel{\sim }{W}},\mu :=\mu \left(H\right),$ where $H:=Re\stackrel{\sim }{W}.$

2.2. Some Facts of the Entire Functions Theory

Here, following the monograph [26], we introduce some notions and facts of the entire function theory. In this subsection, we use the following notations

$$G(z,p):=(1-z)e^{z+\frac{z^2}{2}+\dots +\frac{z^p}{p}},G(z,0):=(1-z).$$

Consider an entire function that has zeros satisfying the following relation for some $\lambda >0$

$$\sum _{n=1}^{\infty }\frac{1}{|a_n{|}^{\lambda }}<\infty .$$

(1)

In this case, we denote by p the smallest integer number for which the following condition holds

$$\sum _{n=1}^{\infty }\frac{1}{|a_n{|}^{p+1}}<\infty .$$

(2)

It is clear that $0\le p<\lambda .$ It is proved that under the assumption (1), the infinite product

$$\prod _{n=1}^{\infty }\left(z\right):=\prod _{n=1}^{\infty }G\left(\frac{z}{a_n},p\right)$$

(3)

is uniformly convergent; we will call it a canonical product and call p the genus of the canonical product. By the convergence exponent $\rho$ of the sequence ${\left\{a_n\right\}}_1^{\infty }\subset \mathbb{C},a_n\ne 0,a_n\to \infty$, we mean the greatest lower bound for such numbers $\lambda$ that the series (1) converges. Note that if $\lambda$ is equal to a convergence exponent, then series (1) may or may not be convergent. For instance, the sequences $a_n=1/n^{\lambda }$ and $1/{(n{ln}^{2}n)}^{\lambda }$ have the same convergence exponent $\lambda =1,$ but in the first case, the series (1) is divergent when $\lambda =1$, while in the second one, it is convergent. In this paper, we have a special interest regarding the first case. Consider the following obvious relation between the convergence exponent $\rho$ and the genus p of the corresponding canonical product $p\le \rho \le p+1.$ It is clear that if $\rho =p,$ then the series (1) diverges for $\lambda =\rho ,$ while $\rho =p+1$ means that the series converges (in accordance with the definition of p). In the monograph [26], a more precise characteristic of the density of the sequence ${\left\{a_n\right\}}_1^{\infty }$ is considered than the convergence exponent. Thus, we defined a so-called counting function $n\left(r\right)$ equal to a number of points of the sequence in the circle $\left|z\right|<r.$ By the upper density of the sequence, we call a number

$$\Delta =\overline{\underset{r\to \infty }{lim}}n\left(r\right)/r^{\rho }.$$

If a limit exists in the ordinary sense (not in the sense of the upper limit), then $\Delta$ is called the density. Note that it is proven in Lemma 1 [26] that

$$\underset{r\to \infty }{lim}n\left(r\right)/r^{\rho +\epsilon }\to 0,\epsilon >0.$$

We need the following fact (see [26] Lemma 3).

Lemma 1.

If the series (2) converges, then the corresponding infinite product (3) satisfies the following inequality in the entire complex plane

$$ln\left|\prod _{n=1}^{\infty }\left(z\right)\right|\le C{r}^p\left(\underset{0}{\overset{r}{\int }}\frac{n\left(t\right)}{t^{p+1}}dt+r\underset{r}{\overset{\infty }{\int }}\frac{n\left(t\right)}{t^{p+2}}dt\right),r:=\left|z\right|.$$

Using this result, it is not hard to prove a relevant fact mentioned in the monograph [26]. Since it has a principal role in the further narrative, then we formulate it as a lemma in terms of the density.

Lemma 2.

Assume that the following series converges for some values $\lambda >0,$ i.e.,

$$\sum _{n=1}^{\infty }\frac{1}{|a_n{|}^{\lambda }}<\infty .$$

Then, the following relation holds

$$\left|\prod _{n=1}^{\infty }\left(z\right)\right|\le e^{\beta \left(r\right)r^{{\rho }_1}},\beta \left(r\right)=r^{p-{\rho }_1}\left(\underset{0}{\overset{r}{\int }}\frac{n\left(t\right)}{t^{p+1}}dt+r\underset{r}{\overset{\infty }{\int }}\frac{n\left(t\right)}{t^{p+2}}dt\right).$$

(4)

In the case ${\rho }_1=\rho ,$ we have $\beta \left(r\right)\to 0,$ if at least one of the following conditions holds: the convergence exponent $\rho <\lambda$ is non-integer and the density equals to zero, or the convergence exponent $\rho =\lambda$ is arbitrary. In addition, the equality $\rho =\lambda$ guarantees that the density equals to zero. In the case ${\rho }_1>\rho ,$ we claim that $\beta \left(r\right)\to 0,$ without any additional conditions.

Proof. 

Applying Lemma 1, we establish relation (4). Consider a case when $\rho <\lambda$ is a non-integer. Taking into account the fact that the density is equal to zero, using L’Hôpital’s rule, we easily obtain

$$r^{p-\rho }\underset{0}{\overset{r}{\int }}\frac{n\left(t\right)}{t^{p+1}}dt\to 0;r^{p+1-\rho }\underset{r}{\overset{\infty }{\int }}\frac{n\left(t\right)}{t^{p+2}}dt\to 0,$$

(5)

(here, we should remark that if $\rho$ is an integer, then $p=\rho$). Therefore, $\beta \left(r\right)\to 0.$ Consider the case that when $\rho =\lambda ,$ then let us rewrite the series (1) in the form of the Stieltjes integral. Then, we have

$$\sum _{n=1}^{\infty }\frac{1}{|a_n{|}^{\lambda }}=\underset{0}{\overset{\infty }{\int }}\frac{dn\left(t\right)}{t^{\rho }}.$$

Using integration by parts formulae, we get

$$\underset{0}{\overset{r}{\int }}\frac{dn\left(t\right)}{t^{\rho }}=\frac{n\left(r\right)}{r^{\rho }}-\frac{n\left(\gamma \right)}{C^{\rho }}+\rho \underset{0}{\overset{r}{\int }}\frac{n\left(t\right)}{t^{\rho +1}}dt.$$

Here, we should note that there exists a neighborhood of the point zero in which $n\left(t\right)=0.$ The latter representation shows us that the following integral converges, i.e.,

$$\underset{0}{\overset{\infty }{\int }}\frac{n\left(t\right)}{t^{\rho +1}}dt<\infty .$$

In its own turn, it follows that

$$\frac{n\left(r\right)}{r^{\rho }}=n\left(r\right)\rho \underset{r}{\overset{\infty }{\int }}\frac{1}{t^{\rho +1}}dt<\rho \underset{r}{\overset{\infty }{\int }}\frac{n\left(t\right)}{t^{\rho +1}}dt\to 0,r\to \infty .$$

Using this fact analogously to the above, applying L’Hôpital’s rule, we conclude that (5) holds if $\rho =\lambda$ is a non-integer. If $\rho =\lambda$ is an integer, then it is clear that we have $\rho =p+1;$ here, we should remind that it is not possible to assume that $\rho =p$ due to the definition of $p.$ In the case $\rho =p+1,$ using the above reasonings, we obtain

$$r^{-1}\underset{0}{\overset{r}{\int }}\frac{n\left(t\right)}{t^{p+1}}dt\to 0;\underset{r}{\overset{\infty }{\int }}\frac{n\left(t\right)}{t^{p+2}}dt\to 0,$$

from what follows the fact that $\beta \left(r\right)\to 0$. The reasonings related to the case ${\rho }_1>\rho$ are absolutely analogous, we left the proof to the reader. The proof is complete. □

Lemma 3.

We claim that the following implication holds

$$lnr\frac{n\left(r\right)}{r^{{\rho }_1}}\to 0,⟹\beta \left(r\right)lnr\to 0,r\to \infty ,$$

where

$$\beta \left(r\right)=r^{p-{\rho }_1}\left(\underset{0}{\overset{r}{\int }}\frac{n\left(t\right)}{t^{p+1}}dt+r\underset{r}{\overset{\infty }{\int }}\frac{n\left(t\right)}{t^{p+2}}dt\right),{\rho }_1\ne p,p+1.$$

Proof. 

Let us define auxiliary functions

$$u_1\left(r\right):=lnr\underset{0}{\overset{r}{\int }}\frac{n\left(t\right)}{t^{p+1}}dt,u_2\left(r\right):=r^{{\rho }_1-p},v_1\left(r\right):=lnr\underset{r}{\overset{\infty }{\int }}\frac{n\left(t\right)}{t^{p+2}}dt,v_2\left(r\right):=r^{{\rho }_1-p-1}.$$

It is clear that

$$u_1^{\prime }\left(r\right):=\frac{1}{r}\underset{0}{\overset{r}{\int }}\frac{n\left(t\right)}{t^{p+1}}dt+lnr\frac{n\left(r\right)}{r^{p+1}};v_1^{\prime }\left(r\right):=\frac{1}{r}\underset{r}{\overset{\infty }{\int }}\frac{n\left(t\right)}{t^{p+2}}dt+lnr\frac{n\left(r\right)}{r^{p+2}}.$$

Therefore,

$$\frac{u_1^{\prime }\left(r\right)}{u_2^{\prime }\left(r\right)}=C{r}^{p-{\rho }_1}\underset{0}{\overset{r}{\int }}\frac{n\left(t\right)}{t^{p+1}}dt+Clnr\frac{n\left(r\right)}{r^{{\rho }_1}};\frac{v_1^{\prime }\left(r\right)}{v_2^{\prime }\left(r\right)}=C{r}^{p+1-{\rho }_1}\underset{r}{\overset{\infty }{\int }}\frac{n\left(t\right)}{t^{p+2}}dt+Clnr\frac{n\left(r\right)}{r^{{\rho }_1}}.$$

(6)

Notice that $\beta \left(r\right)lnr=u_1\left(r\right)/u_2\left(r\right)+v_1\left(r\right)/v_2\left(r\right),$ applying L’Hôpital’s rule, we have

$$\beta \left(r\right)lnr\sim \frac{u_1^{\prime }\left(r\right)}{u_2^{\prime }\left(r\right)}+\frac{v_1^{\prime }\left(r\right)}{v_2^{\prime }\left(r\right)},r\to \infty .$$

(7)

In an analogous way, we obtain the following implication

$$\frac{n\left(r\right)}{r^{{\rho }_1}}\to 0,⟹\left\{r^{p-{\rho }_1}\underset{0}{\overset{r}{\int }}\frac{n\left(t\right)}{t^{p+1}}dt\to 0;r^{p+1-{\rho }_1}\underset{r}{\overset{\infty }{\int }}\frac{n\left(t\right)}{t^{p+2}}dt\to 0\right\}.$$

(8)

Thus, taking into account the condition $lnr·n\left(r\right)/r^{{\rho }_1}\to 0,$ combining (6)–(8), we obtain the desired result. □

Regarding Lemma 3, we can produce the following example that indicates the relevance of the issue itself.

Example 1.

There exists a sequence ${\left\{a_n\right\}}_1^{\infty }$ such that the density is equal to zero; moreover,

$$\sum _{n=1}^{\infty }\frac{1}{|a_n{|}^{\rho }}=\infty ,\beta \left(r\right)lnr\to 0,r\to \infty ,$$

where

$$\beta \left(r\right)=r^{p-\rho }\left(\underset{0}{\overset{r}{\int }}\frac{n\left(t\right)}{t^{p+1}}dt+r\underset{r}{\overset{\infty }{\int }}\frac{n\left(t\right)}{t^{p+2}}dt\right).$$

We can construct the required sequence supposing $n\left(r\right)\sim r^{{\rho }_1}{(lnr·lnlnr)}^{-1},{\rho }_1\in {\mathbb{R}}^{+}\backslash \mathbb{N}.$

It is clear that we can represent partial sums of series (1) due to the Stieltjes integral

$$\sum _{n=1}^k\frac{1}{|a_n{|}^{\lambda }}=\underset{0}{\overset{r\left(k\right)}{\int }}\frac{dn\left(t\right)}{t^{\lambda }},\lambda \ge {\rho }_1.$$

Thus, the sequence ${\left\{a_n\right\}}_1^{\infty }$ is defined by the function $n\left(r\right).$ Applying the integration by parts formulae, we obtain

$$\underset{0}{\overset{r}{\int }}\frac{dn\left(t\right)}{t^{\lambda }}=\frac{n\left(r\right)}{r^{\lambda }}-\frac{1}{|a_1{|}^{\lambda }}+\lambda \underset{0}{\overset{r}{\int }}\frac{n\left(t\right)}{t^{\lambda +1}}dt.$$

We can easily establish the fact that the last integral diverges when $\lambda ={\rho }_1,r\to \infty ;$ then, we have

$$\underset{0}{\overset{r}{\int }}\frac{n\left(t\right)}{t^{{\rho }_1+1}}dt\ge C\underset{|a_1|}{\overset{r}{\int }}\frac{dt}{tlnt·lnlnt}=lnlnlnr-C.$$

On the other hand, we have

$$\underset{0}{\overset{\infty }{\int }}\frac{n\left(t\right)}{t^{\lambda +1}}dt\le C\underset{|a_1|}{\overset{\infty }{\int }}\frac{dt}{t^{1+\lambda -{\rho }_1}lnt·lnlnt}<\infty ,\lambda >{\rho }_1.$$

Taking into account the fact $n\left(r\right)/r^{{\rho }_1}\to 0,r\to \infty ,$ we conclude that series (1) diverges if $\lambda ={\rho }_1$ and converges if $\lambda ={\rho }_1+\epsilon ,\epsilon >0.$ Therefore, the convergence exponent is equal to ${\rho }_1,$ i.e., $\rho ={\rho }_1,$ the density is equal to zero. Let us prove the fact $\beta \left(r\right)lnr\to 0,$ for this purpose, in accordance with Lemma 3, it suffices to show that

$$lnr\frac{n\left(r\right)}{r^{\rho }}\to 0,r\to \infty ,$$

by direct substitution, we obtain

$$lnr\frac{n\left(r\right)}{r^{\rho }}\le C{(lnlnr)}^{-1}\to 0,r\to \infty .$$

Thus, we obtain the desired result.

2.3. Schatten–Von Neumann Class and the Particular Case Corresponding to the Normal Operator

Let ${\mathfrak{S}}_q\left(\mathfrak{H}\right),0<q<\infty$ be a Schatten–von Neumann class and ${\mathfrak{S}}_{\infty }\left(\mathfrak{H}\right)$ be the set of compact operators. By definition, put

$${\mathfrak{S}}_q\left(\mathfrak{H}\right):=\left\{T:\mathfrak{H}\to \mathfrak{H},\sum _{i=1}^{\infty }{s}_i^q\left(L\right)<\infty ,0<q<\infty \right\}.$$

Denote by ${\stackrel{\sim }{\mathfrak{S}}}_{\rho }\left(\mathfrak{H}\right)$ the class of the operators such that

$${\stackrel{\sim }{\mathfrak{S}}}_{\rho }\left(\mathfrak{H}\right):=\{T\in {\mathfrak{S}}_{\rho +\epsilon },T\overline{\in }{\mathfrak{S}}_{\rho -\epsilon },\forall \epsilon >0\},$$

We will call it the Schatten–von Neumann class of the convergence exponent. Note that there exists a one-to-one correspondence between selfadjoint compact operators and monotonically decreasing sequences. If we consider Example 1, then we see that the made definition becomes relevant in this regard.

Lemma 4.

Assume that

$${\left({ln}^{\kappa +1}x\right)}_{{\lambda }_i\left(H\right)}^{\prime }=o\left(i^{-\kappa }\right),\kappa \in (0,1].$$

Then, in the general case, we obtain

$$A\in {\stackrel{\sim }{\mathfrak{S}}}_{\rho },\rho \in [0,2/\kappa ],n_A\left(r\right)=o(r^{2/\kappa }/lnr).$$

In the particular case, when $\stackrel{\sim }{W}$ is normal, we obtain

$$A\in {\stackrel{\sim }{\mathfrak{S}}}_{\rho },\rho \in [0,1/\kappa ],n_A\left(r\right)=o(r^{1/\kappa }/lnr).$$

Moreover, the additional assumption ${\lambda }_i\left(H\right)=O\left(i^{\kappa +\epsilon }\right),\forall \epsilon >0$ gives us the estimate $\rho \ge 1/\kappa$ in both cases; thus, in the case when $\stackrel{\sim }{W}$ is normal, we obtain

$$A\in {\stackrel{\sim }{\mathfrak{S}}}_{1/\kappa }.$$

Proof. 

Note that the fact $A\in {\stackrel{\sim }{\mathfrak{S}}}_{\rho },\rho \in [0,2/\kappa ],$ follows directly from Theorem 1, claim $\left(\mathbf{A}\right).$ In accordance with relation (54) [4], we have ${\left(\right|A|}^2{f,f)}_{\mathfrak{H}}={\parallel Af\parallel }_{\mathfrak{H}}^2\le C·\mathrm{Re}{(Af,f)}_{\mathfrak{H}}=C{(Vf,f)}_{\mathfrak{H}},$ where $V:=(A+A^{\ast })/2.$ Applying Theorem 5 [4], we obtain ${\lambda }_i\left(V\right)\asymp {\lambda }_i\left(R_H\right);$ thus, we obtain $s_i\left(A\right)\le C{\lambda }_i^{1/2}\left(R_H\right);s_i^{-1}\left(A\right)\ge C{\lambda }_i^{1/2}\left(H\right).$ The detailed proof of the latter fact can be seen in Theorem 7 [4]. Using the monotonous property of the functions, we have

$$\frac{{ln}^{\kappa }{s}_i^{-1}\left(A\right)}{s_i^{-2}\left(A\right)}\le C·\frac{{ln}^{\kappa }{\lambda }_i\left(H\right)}{{\lambda }_i\left(H\right)}\le C·\frac{{\alpha }_i}{i^{\kappa }},$$

where ${\alpha }_i\to 0.$ Hence,

$$\frac{iln{s}_i^{-1}\left(A\right)}{s_i^{-2/\kappa }\left(A\right)}\le C·{\alpha }_i^{1/\kappa }.$$

Taking into account the facts $n\left(s_i^{-1}\right)=i;n\left(r\right)=n\left(s_i^{-1}\right),s_i^{-1}<r<s_{i+1}^{-1},$ using the monotonous property of the functions, we obtain

$$\frac{n\left(r\right)lnr}{r^{2/\kappa }}<C·{\alpha }_i,s_i^{-1}<r<s_{i+1}^{-1}.$$

The proof corresponding to the general case is complete. Assume that the operator $\stackrel{\sim }{W}$ is normal; then, it is not hard to prove that A is normal also. Let us show that the operator $V:=(A+A^{\ast })/2$ has a complete orthonormal system of the eigenvectors. Using formula (53) [4], we obtain

$$V^{{}^{-1}}=2H^{\frac{1}{2}}(I+B^2)H^{\frac{1}{2}}.$$

Note that in accordance with relation (67) [4], we have

$${(V^{{}^{-1}}f,f)}_{\mathfrak{H}}=2{(S{H}^{\frac{1}{2}}f,H^{\frac{1}{2}}f)}_{\mathfrak{H}}\ge 2{\parallel H^{\frac{1}{2}}f\parallel }_{\mathfrak{H}}^2=2{(Hf,f)}_{\mathfrak{H}},f\in \mathrm{D}(V^{{}^{-1}}),$$

(9)

where $S=I+B^2.$ Since V is selfadjoint, then due to Theorem 3 [27] p. 136, the operator $V^{{}^{-1}}$ is selfadjoint also. Combining (9) with Lemma 3 [4], we get that $V^{{}^{-1}}$ is strictly accretive. Using these facts, we can write

$${\parallel f\parallel }_{V^{{}^{-1}}}\ge C{\parallel f\parallel }_H,f\in {\mathfrak{H}}_{V^{{}^{-1}}},$$

where the above norms are understood as the norms of the energetic spaces generated by the operators $V^{{}^{-1}}$ and H, respectively. Since the operator H has a discrete spectrum (see Theorem 5.3 [28]), then any set bounded with respect to the norm ${\mathfrak{H}}_H$ is a compact set with respect to the norm $\mathfrak{H}$ (see Theorem 4 [29] p. 220). Combining this fact with (9), Theorem 3 [29] p. 216, we obtain that the operator $V^{{}^{-1}}$ has a discrete spectrum, i.e., it has the infinite set of the eigenvalues ${\lambda }_1\le {\lambda }_2\le \dots \le {\lambda }_i\le \dots ,{\lambda }_i\to \infty ,i\to \infty$ and the complete orthonormal system of the eigenvectors. Now, note that the operators $V,V^{{}^{-1}}$ have the same eigenvectors. Therefore, the operator V has the complete orthonormal system of the eigenvectors. Recall that any complete orthonormal system forms a basis in the separable Hilbert space. Hence, the complete orthonormal system of the eigenvectors of the operator V is a basis in the space $\mathfrak{H}.$ Since the operator A is compact and normal, then in accordance with the well-known theorem, we have a fact that there exists an orthonormal system of the eigenvectors ${\left\{{\psi }_i\right\}}_1^{\infty }$ of the operator $A.$ The system is complete in $\overline{\mathrm{R}\left(A\right)}$ in the following sense

$$f=\sum _{i=1}^{\infty }{\psi }_i{(f,{\psi }_i)}_{\mathfrak{H}},f\in \overline{\mathrm{R}\left(A\right)}.$$

The corresponding system of eigenvalues is such that

$$A{\psi }_i={\lambda }_i\left(A\right){\psi }_i,A^{\ast }{\psi }_i=\overline{{\lambda }_i\left(A\right)}{\psi }_i,i\in \mathbb{N}.$$

The latter facts give us $A^{\ast }A{\psi }_i={\left|{\lambda }_i\left(A\right)\right|}^2{\psi }_i.$ Since the operator $A^{\ast }A$ is selfadjoint and compact, then it is not hard to prove that $s_i\left(A\right)=\left|{\lambda }_i\left(A\right)\right|$ (see Lemma 3.3 Chapter II [2]). Thus, we obtain

$$s_i\left(A\right)=|{(A{\psi }_i,{\psi }_i)}_{\mathfrak{H}}|=\left(1+{tan}^2{\theta }_i\right)\left|\mathrm{Re}{(A{\psi }_i,{\psi }_i)}_{\mathfrak{H}}\right|=$$

(10)

$$=\left(1+{tan}^2{\theta }_i\right)\left|{(V{\psi }_i,{\psi }_i)}_{\mathfrak{H}}\right|=\left(1+{tan}^2{\theta }_i\right){\lambda }_i\left(V\right),$$

where the sequence ${\left\{{tan}^2{\theta }_i\right\}}_1^{\infty }$ is bounded by virtue of the sectorial property of the operator. Note that the fact $\overline{\mathrm{R}\left(A\right)}=\mathfrak{H}$ indicates that ${\left\{{\psi }_i\right\}}_1^{\infty }$ is complete in $\mathfrak{H}.$ It follows that the operators V and A have the same eigenvectors (since the complete system of the eigenvectors of the operator V is minimal and at the same time, it contains all eigenvectors of the operator A). Therefore, we can claim that all eigenvalues of the operator V are involved in the right-hand side of relation (10). Taking into account the fact that ${\lambda }_i\left(V\right)\asymp {\lambda }_i\left(R_H\right),$ we obtain the following relation

$$\sum _{i=1}^{\infty }|s_i{\left(A\right)|}^p\le C_2\sum _{i=1}^{\infty }{|{\lambda }_i\left(R_H\right)|}^p,p>0.$$

Using the theorem condition, we have $A\in {\mathfrak{S}}_p,p>1/\kappa .$ Hence, $A\in {\stackrel{\sim }{\mathfrak{S}}}_{\rho },\rho \le 1/\kappa .$ At the same time, applying the above reasonings, we obtain

$$\frac{{ln}^{\kappa }{s}_i^{-1}\left(A\right)}{s_i^{-1}\left(A\right)}\le C·\frac{{ln}^{\kappa }{\lambda }_i\left(H\right)}{{\lambda }_i\left(H\right)}\le C·\frac{{\alpha }_i}{i^{\kappa }}.$$

Using this fact, we obtain

$$\frac{n\left(r\right)lnr}{r^{1/\kappa }}\to 0,r\to \infty .$$

Consider the additional condition ${\lambda }_i\left(H\right)=O\left(i^{\kappa +\epsilon }\right),\forall \epsilon >0$ and let ${\left\{{\psi }_i\right\}}_1^{\infty }$ still be the complete orthonormal system of the eigenvectors of the operator $V.$ Suppose $A\in {\mathfrak{S}}_p,p\ge 1;$ then, by virtue of inequalities (7.9) Chapter III [2], the fact ${\lambda }_i\left(V\right)\asymp {\lambda }_i\left(R_H\right)$ (see Theorem 5 [28]), we obtain

$$\sum _{i=1}^{\infty }|s_i{\left(A\right)|}^p\ge \sum _{i=1}^{\infty }|{(A{\phi }_i,{\phi }_i)}_{\mathfrak{H}}{|}^p\ge \sum _{i=1}^{\infty }{|\mathrm{Re}{(A{\phi }_i,{\phi }_i)}_{\mathfrak{H}}|}^p$$

$$=\sum _{i=1}^{\infty }|{(V{\phi }_i,{\phi }_i)}_{\mathfrak{H}}{|}^p=\sum _{i=1}^{\infty }{|{\lambda }_i\left(V\right)|}^p\ge C\sum _{i=1}^{\infty }{i}^{-(\kappa +\epsilon )p}.$$

Therefore, $A\in {\mathfrak{S}}_{\rho },\rho \ge 1/\kappa$, since in the contrary case, the relation $p(\kappa +\epsilon )>1$ does not hold. The proof is complete. □

Consider the following example.

Example 2.

Here, we would like to produce an example of the sequence ${\left\{{\lambda }_i\right\}}_1^{\infty }$ that satisfies the condition

$${\left({ln}^{\kappa +1}x\right)}_{{\lambda }_i}^{\prime }=o\left(i^{-\kappa }\right),\kappa \in (0,1],$$

$$\sum _{n=1}^{\infty }\frac{1}{|{\lambda }_n{|}^{1/\kappa }}=\infty .$$

Consider a sequence ${\lambda }_i=i^{\kappa }{ln}^{\kappa }(i+q)·{ln}^{\kappa }ln(i+q),q>e^e-1,i=1,2,\dots$. Using the integral test for convergence, we can easily see that the previous series is divergent. At the same time, substituting, we obtain

$$\frac{{ln}^{\kappa }{\lambda }_i}{{\lambda }_i}\le \frac{C{ln}^{\kappa }(i+q)}{i^{\kappa }{ln}^{\kappa }(i+q)·{ln}^{\kappa }ln(i+q)}=\frac{C}{i^{\kappa }·{ln}^{\kappa }ln(i+q)},$$

which gives us the fulfilment of the condition.

Below, we produce an auxiliary technique to study the central problem of the paper. The estimates for the Fredholm Determinant were studied by Lidskii in the paper [3] and gave a main tool for solving the problems related to the estimation of the contour integrals. We have slightly improved results by Lidskii having involved the auxiliary function $\beta$ and obtaining in this way more accurate results.

2.4. Estimates for the Fredholm Determinant

In this section, we produce an adopted version of the propositions given in the paper [3]; we consider a case when a compact operator belongs to the class ${\stackrel{\sim }{\mathfrak{S}}}_{\rho }.$ Having taken into account the facts considered in the previous subsection, we can reformulate Lemma 2 [3] in the refined form.

Lemma 5.

Assume that a compact operator B satisfies the condition $B\in {\stackrel{\sim }{\mathfrak{S}}}_{\rho };$ then, for arbitrary numbers $R,\delta$ such that $R>0,0<\delta <1,$ there exists a circle $\left|\lambda \right|=\stackrel{\sim }{R},(1-\delta )R<\stackrel{\sim }{R}<R,$ so that the following estimate holds

$${\parallel (I-\lambda B)}^{-1}{\parallel }_{\mathfrak{H}}\le e^{{\gamma \left(\right|\lambda \left|\right)\left|\lambda \right|}^{\varrho }}{\left|\lambda \right|}^m,\left|\lambda \right|=\stackrel{\sim }{R},m=\left[\varrho \right],\varrho \ge \rho ,$$

where

$$\gamma \left(\right|\lambda \left|\right)={\beta \left(\right|\lambda |}^{m+1}{)+C\beta (\left|C\lambda \right|}^{m+1}),\beta \left(r\right)=r^{-\frac{\varrho }{m+1}}\left(\underset{0}{\overset{r}{\int }}\frac{n_{B^{m+1}}\left(t\right)dt}{t}+r\underset{r}{\overset{\infty }{\int }}\frac{n_{B^{m+1}}\left(t\right)dt}{t^2}\right).$$

Proof. 

We consider the case $\varrho =\rho ,$ the reasonings corresponding to the case $\varrho >\rho$ can be fulfilled in accordance with the same scheme but much easier; we left them to the reader. Using the definition, we have $B\in {\mathfrak{S}}_{\rho +\epsilon },\epsilon >0.$ By direct calculation, we obtain

$${(I-{\lambda }^{m+1}{B}^{m+1})}^{-1}(I+\lambda B+{\lambda }^2{B}^2+\dots +{\lambda }^m{B}^m)={(I-\lambda B)}^{-1}.$$

(11)

In accordance with Lemma 3 [3], for sufficiently small $\epsilon >0,$ we have

$$\sum _{i=1}^{\infty }{\lambda }_i^{\frac{\rho +\epsilon }{m+1}}(\stackrel{\sim }{B})\le \sum _{i=1}^{\infty }{\lambda }_i^{\rho +\epsilon }\left(B\right)<\infty ,$$

where $\stackrel{\sim }{B}:={\left(B^{\ast m+1}{B}^{m+1}\right)}^{1/2}.$ Applying inequality (1.27) [3] p. 10, (since $\rho /(m+1)<1$), using Lemma 2, we obtain

$$\parallel {\Delta }_{B^{m+1}}\left({\lambda }^{m+1}\right){(I-{\lambda }^{m+1}{B}^{m+1})}^{-1}{\parallel }_{\mathfrak{H}}\le C\prod _{i=1}^{\infty }\{1+|{\lambda }^{m+1}{s}_i\left(B^{m+1}\right)\left|\right\}\le C{e}^{\beta \left(r^{m+1}\right)r^{\rho }},$$

where ${\Delta }_{B^{m+1}}\left({\lambda }^{m+1}\right)$ is a Fredholm determinant of the operator $B^{m+1}$ (see [3] p. 8). In accordance with Theorem 11 [26] p. 33, we have

$${\Delta }_{B^{m+1}}\left({\lambda }^{m+1}\right)\ge e^{-(2+ln\{12e/\delta \})ln{\xi }_m},{\xi }_m=\underset{\psi \in [0,2\pi /(m+1\left)\right]}{max}\{{\Delta }_{B^{m+1}}({[2e\stackrel{\sim }{R}{e}^{i\psi }]}^{m+1})\},$$

where $R,\delta$ are arbitrary numbers such that $R>0,0<\delta <1,$ the values of $\lambda$ belong to the circle $\left|\lambda \right|=\stackrel{\sim }{R},$ whose radius is defined by $R,\delta$ and satisfies the condition $(1-\delta )R<\stackrel{\sim }{R}<R.$ Note that in accordance with the estimate (1.21) [3] p. 10, we have

$${\Delta }_{B^{m+1}}\left(\lambda \right)\le C\prod _{i=1}^{\infty }\{1+|\lambda s_i\left(B^{m+1}\right)\left|\right\}.$$

Therefore, applying Lemma 2, we obtain ${\xi }_m\le e^{\beta ({[2e\stackrel{\sim }{R}]}^{m+1}){(2e\stackrel{\sim }{R})}^{\rho }}$. Consider relation (11), we have

$${\parallel (I-\lambda B)}^{-1}{\parallel }_{\mathfrak{H}}\le \parallel {(I-{\lambda }^{m+1}{B}^{m+1})}^{-1}{\parallel }_{\mathfrak{H}}·{\parallel (I+\lambda B+{\lambda }^2{B}^2+\dots +{\lambda }^m{B}^m)\parallel }_{\mathfrak{H}}$$

$$\le \parallel {(I-{\lambda }^{m+1}{B}^{m+1})}^{-1}{\parallel }_{\mathfrak{H}}·\frac{{\left|\lambda \right|}^{m+1}{\parallel B\parallel }^{m+1}-1}{\left|\lambda \right|·\parallel B\parallel -1}.$$

We can easily see that to obtain the desired result, it suffices to estimate the term $\parallel {(I-{\lambda }^{m+1}{B}^{m+1})}^{-1}{\parallel }_{\mathfrak{H}}.$ Using the obtained estimates, we have

$$\parallel {(I-{\lambda }^{m+1}{B}^{m+1})}^{-1}{\parallel }_{\mathfrak{H}}\le e^{{\gamma \left(\right|\lambda \left|\right)\left|\lambda \right|}^{\rho }},\left|\lambda \right|=\stackrel{\sim }{R},$$

where $\gamma \left(\right|\lambda \left|\right)={\beta \left(\right|\lambda |}^{m+1})+(2+ln\{12e/\delta \}){\beta }_m{\left(\right|2e\lambda |}^{m+1}{)\left(2e\right)}^{\rho }.$ Thus, we obtain the desired result. □

2.5. Abel–Lidsky Summarizing the Series

In this subsection, we reformulate results obtained by Lidskii [3] in a more convenient form applicable to the reasonings of this paper. However, let us begin our narrative. In accordance with the Hilbert theorem (see [30] and (p. 32, [2])), the spectrum of an arbitrary compact operator B consists of the so-called normal eigenvalues; this gives us an opportunity to consider a decomposition to a direct sum of subspaces

$$\mathfrak{H}={\mathfrak{N}}_q\oplus {\mathfrak{M}}_q,$$

(12)

where both summands are invariant subspaces regarding the operator $B$. The first one is a finite dimensional root subspace corresponding to the eigenvalue ${\mu }_q$, and the second one is a subspace wherein the operator $B-{\mu }_{q}I$ is invertible. Let $n_q$ be a dimension of ${\mathfrak{N}}_q$ and let $B_q$ be the operator induced in ${\mathfrak{N}}_q$. We can choose a basis (Jordan basis) in ${\mathfrak{N}}_q$ that consists of Jordan chains of eigenvectors and root vectors of the operator $B_q$. Each chain $e_{q_{\xi }},e_{q_{\xi }+1},\dots ,e_{q_{\xi }+k},k\in {\mathbb{N}}_0,$ where $e_{q_{\xi }},\xi =1,2,\dots ,m$ are the eigenvectors corresponding to the eigenvalue ${\mu }_q$ and other terms are root vectors; these can be transformed by the operator B in accordance with the following formulas

$$B{e}_{q_{\xi }}={\mu }_q{e}_{q_{\xi }},B{e}_{q_{\xi }+1}={\mu }_q{e}_{q_{\xi }+1}+e_{q_{\xi }},\dots ,B{e}_{q_{\xi }+k}={\mu }_q{e}_{q_{\xi }+k}+e_{q_{\xi }+k-1}.$$

(13)

Considering the sequence ${\left\{{\mu }_q\right\}}_1^{\infty }$ of the eigenvalues of the operator B and choosing a Jordan basis in each corresponding space ${\mathfrak{N}}_q,$ we can arrange a system of vectors ${\left\{e_i\right\}}_1^{\infty }$ which we will call a system of the root vectors, or following Lidskii, a system of the major vectors of the operator $B$. Assume that $e_1,e_2,\dots ,e_{n_q}$ is the Jordan basis in the subspace ${\mathfrak{N}}_q,$ let us prove that (see [3] p. 14) there exists a corresponding biorthogonal basis $g_1,g_2,\dots ,g_{n_q}$ in the subspace ${\mathfrak{M}}_q^{\perp }$. It is easy to prove that the subspace ${\mathfrak{M}}_q^{\perp }$ has the same dimension equal to $n_q.$ For this purpose, assume that the vectors $f_j\in {\mathfrak{M}}_q^{\perp },j=1,2,\dots ,l$ are linearly independent; then, using the decomposition of the space $\mathfrak{H}$ to the direct sum $f_j=a_j+b_j,a_j\in {\mathfrak{N}}_q,b_j\in {\mathfrak{M}}_q,$ we obtain

$$\sum _{j=1}^l{f}_j{\alpha }_j=\sum _{j=1}^l{a}_j{\alpha }_j+\sum _{j=1}^l{b}_j{\alpha }_j\ne 0,$$

where ${\left\{{\alpha }_j\right\}}_1^l\subset \mathbb{C}$ is an arbitrary non-zero set. It implies that

$$\sum _{j=1}^l{a}_j{\alpha }_j\ne 0,$$

for if we assume the contrary, then we will come to the contradiction. Hence, $dim{\mathfrak{M}}_q^{\perp }\le dim{\mathfrak{N}}_q.$ Using the same reasonings, we obtain the fact that $dim{\mathfrak{M}}_q^{\perp }\ge dim{\mathfrak{N}}_q.$ Thus, we obtain the desired result. Now, let us choose an element $e_p\in {\left\{e_i\right\}}_1^{n_q}$ and consider $(n_q-1)$—dimensional space ${\mathfrak{N}}_q^{\left(p\right)}$ generated by the set ${\left\{e_i\right\}}_1^{n_q}\backslash e_p;$ then, let us choose an arbitrary element $g_p\ne 0$ belonging to the orthogonal complement of the set ${\mathfrak{N}}_q^{\left(p\right)}\oplus {\mathfrak{M}}_q.$ It is clear that $g_p\in {\mathfrak{M}}_q^{\perp }$, since in accordance with the given definition, the element $g_p$ is orthogonal to the set ${\mathfrak{M}}_q.$ It is also clear that in accordance with the definition ${(e_j,g_p)}_{\mathfrak{H}}=0,j\ne p.$ Note that ${(e_p,g_p)}_{\mathfrak{H}}\ne 0,$ for if we assume the contrary, then using the decomposition (12), we obtain ${(g_p,f)}_{\mathfrak{H}}=0,f\in \mathfrak{H}$, and as a result, we obtain the contradiction, i.e., $g_p=0.$ It is clear that we can choose $g_p$ so that ${(e_p,g_p)}_{\mathfrak{H}}=1.$ Let us show that the system of the elements constructed in this way ${\left\{g_i\right\}}_1^{n_q}$ is linearly independent. It follows easily from the implication

$$\sum _{j=1}^{n_q}{g}_j{\alpha }_j=0,\Rightarrow \sum _{j=1}^{n_q}{\alpha }_j(g_j,e_p)=0,\Rightarrow {\alpha }_p=0,p\in \{1,2,\dots ,n_q\}.$$

Therefore, taking into account the proved above fact $dim{\mathfrak{M}}_q^{\perp }=n_q,$ we conclude that the system ${\left\{g_i\right\}}_1^{n_q}$ is a basis in ${\mathfrak{M}}_q^{\perp }.$ Let us show that the system ${\left\{g_i\right\}}_1^{n_q}$ consists of the Jordan chains of the operator $B^{\ast }$, which correspond to the Jordan chains (13). Note that the space ${\mathfrak{M}}_q^{\perp }$ is an invariant subspace of the operator $B^{\ast }$, since it is orthogonal to the invariant subspace of the operator $B.$ Using the denotation $B^{\left(q\right)}$ for the operator B restriction on the invariant subspace ${\mathfrak{N}}_q,$ let us denote by $B^{\left(q\right)\ast }$ a restriction of the operator $B^{\ast }$ on the subspace ${\mathfrak{M}}_q^{\perp }.$ Assume that $\mathrm{B}\left({\mu }_q\right)$ is a matrix of the operator $B^{\left(q\right)}$ in the basis ${\left\{e_i\right\}}_1^{n_q};$ then, using conditions (13), we conclude that it has a Jordan form, i.e., it is a block diagonal matrix, where each Jordan block is represented by a matrix in the normal Jordan form, i.e.,

$$\mathrm{B}\left({\mu }_q\right)=\left(\begin{array}{cccccc}{\mathrm{b}}_{q_1}\left({\mu }_q\right)& \dots & \dots & \dots & \dots & \dots \\ \dots & {\mathrm{b}}_{q_2}\left({\mu }_q\right)& \dots & \dots & \dots & \dots \\ \dots & \dots & {\mathrm{b}}_{q_3}\left({\mu }_q\right)& \dots & \dots & \dots \\ \dots \\ \\ \dots & \dots & \dots & \dots & \dots & {\mathrm{b}}_{q_m}\left({\mu }_q\right)\end{array}\right),{\mathrm{b}}_{q_{\xi }}\left({\mu }_q\right)=\left(\begin{array}{cccccc}{\mu }_q& 1& 0& 0& 0& \dots \\ 0& {\mu }_q& 1& 0& 0& \dots \\ 0& 0& {\mu }_q& 1& 0& \dots \\ \dots \\ \\ 0& \dots & \dots & 0& 0& {\mu }_q\end{array}\right),$$

$$dim{\mathrm{b}}_{q_{\xi }}\left({\mu }_q\right)=k\left(q_{\xi }\right)+1,\xi =1,2,\dots ,m,$$

where $m:=m\left(q\right)$ is a geometrical multiplicity of the q-th eigenvalue, and $k\left(q_{\xi }\right)+1$ is a number of elements in the $q_{\xi }$-th Jordan chain. Since we have $B^{\left(q\right)}:{\mathfrak{N}}_q\to {\mathfrak{N}}_q,B^{\left(q\right)\ast }:{\mathfrak{M}}_q^{\perp }\to {\mathfrak{M}}_q^{\perp },$ then

$$B^{\left(q\right)}{e}_i=\sum _{j=1}^{n_q}{\alpha }_{ji}{e}_j,B^{\left(q\right)\ast }{g}_j=\sum _{i=1}^{n_q}{\gamma }_{ij}{e}_i,$$

where $\left\{{\alpha }_{ji}\right\},\left\{{\gamma }_{ij}\right\},i,j=1,2,\dots ,n_q$ are the matrices of the operators $B^{\left(q\right)},B^{\left(q\right)\ast }$ in the bases ${\left\{e_i\right\}}_1^{n_q},{\left\{g_i\right\}}_1^{n_q}$, respectively. On the other hand, we have the obvious reasonings

$${\alpha }_{ji}={(B^{\left(q\right)}{e}_i,e_j)}_{\mathfrak{H}}={(B{e}_i,e_j)}_{\mathfrak{H}}={(e_i,B^{\ast }{e}_j)}_{\mathfrak{H}}={(e_i,B^{\left(q\right)\ast }{e}_j)}_{\mathfrak{H}}={\overline{\gamma }}_{ij}.$$

Therefore, we conclude that the operator $B^{\left(q\right)\ast }$ is represented by a matrix

$${\mathrm{B}}^{\top }\left({\overline{\mu }}_q\right)=\left(\begin{array}{cccccc}{\mathrm{b}}_{q_1}^{\top }\left({\overline{\mu }}_q\right)& \dots & \dots & \dots & \dots & \dots \\ \dots & {\mathrm{b}}_{q_2}^{\top }\left({\overline{\mu }}_q\right)& \dots & \dots & \dots & \dots \\ \dots & \dots & {\mathrm{b}}_{q_3}^{\top }\left({\overline{\mu }}_q\right)& \dots & \dots & \dots \\ \dots \\ \\ \dots & \dots & \dots & \dots & \dots & {\mathrm{b}}_{q_m}^{\top }\left({\overline{\mu }}_q\right)\end{array}\right).$$

Using this representation, we conclude that ${\left\{g_i\right\}}_1^{n_q}$ consists of the Jordan chains of the operator $B^{\ast }$, which correspond to the Jordan chains (13) due to the following formula

$$B^{\ast }{g}_{q_{\xi }+k}={\overline{\mu }}_q{g}_{q_{\xi }+k},B^{\ast }{g}_{q_{\xi }+k-1}={\overline{\mu }}_q{g}_{q_{\xi }+k-1}+g_{q_{\xi }+k},\dots ,B^{\ast }{g}_{q_{\xi }}={\overline{\mu }}_q{g}_{q_{\xi }}+g_{q_{\xi }+1}.$$

Let us show that ${\mathfrak{N}}_i\subset {\mathfrak{M}}_j;i\ne j$ for this purpose, note that in accordance with the property $P_{{\mu }_i}{P}_{{\mu }_j}=0,i\ne j,$ where $P_{{\mu }_i}$ is a Riesz projector (integral) corresponding to the eigenvalue ${\mu }_i$ (see [2] Chapter I §1.3), and the property $P_{{\mu }_i}f=f,f\in {\mathfrak{N}}_i,$ we have

$$P_{{\mu }_j}f=P_{{\mu }_j}{P}_{{\mu }_i}f=0,f\in {\mathfrak{N}}_i.$$

Combining this relation with the decomposition (12), we obtain the desired result. Now, taking into account relation (12), we conclude that the set ${\left\{g_{\nu }\right\}}_1^{n_j},j\ne i$ is orthogonal to the set ${\left\{e_{\nu }\right\}}_1^{n_i}.$ Gathering the sets ${\left\{g_{\nu }\right\}}_1^{n_j},j=1,2,\dots ,$ we can obviously create a biorthogonal system ${\left\{g_n\right\}}_1^{\infty }$ with respect to the system of the major vectors of the operator $B.$ It is rather reasonable to call it a system of the major vectors of the operator $B^{\ast }.$ Note that if an element $f\in \mathfrak{H}$ allows a decomposition in the strong sense

$$f=\sum _{n=1}^{\infty }{e}_n{c}_n,c_n\in \mathbb{C},$$

then by virtue of the biorthogonal system existing, we can claim that such a representation is unique. Furthermore, let us come to the previously made agreement that the vectors in each Jordan chain are arranged in the same order as in (13), i.e., at the first place, there stands an eigenvector. It is clear that under such an assumption, we have

$$c_{q_{\xi }+i}=\frac{(f,g_{q_{\xi }+k-i})}{(e_{q_{\xi }+i},g_{q_{\xi }+k-i})},0\le i\le k\left(q_{\xi }\right),$$

where $k\left(q_{\xi }\right)+1$ is a number of elements in the $q_{\xi }$-th Jourdan chain. In particular, if the vector $e_{q_{\xi }}$ is included to the major system solo, there does not exist a root vector corresponding to the corresponding eigenvalue, then

$$c_{q_{\xi }}=\frac{(f,g_{q_{\xi }})}{(e_{q_{\xi }},g_{q_{\xi }})}.$$

Note that in accordance with the property of the biorthogonal sequences, we can expect that the denominators are equal to one in the previous two relations. Consider a formal series corresponding to a decomposition on the major vectors of the operator B

$$f\sim \sum _{n=1}^{\infty }{e}_n{c}_n,$$

where each number n corresponds to a number $q_{\xi }+i$ (thus, the coefficients $c_n$ are defined in accordance with the above and numerated in a simplest way). Consider a set of the polynomials with respect to a real parameter t

$$P_{\nu }^{\alpha }({\zeta }^{-1},t)=\frac{e^{t{\zeta }^{-\alpha }}}{\nu !}\frac{d^{\nu }}{d{\zeta }^{\nu }}{e}^{-t{\zeta }^{-\alpha }},\alpha >0,\nu =1,2,\dots .$$

Consider a series

$$\sum _{n=1}^{\infty }{c}_n\left(t\right)e_n,$$

(14)

where the coefficients $c_n\left(t\right)$ are defined in accordance with the correspondence between the indexes n and $q_{\xi }+i$ in the following way

$$c_{q_{\xi }+i}\left(t\right)=e^{-{\lambda }_q^{\alpha }t}\sum _{\nu =0}^{k-i}{P}_{\nu }^{\alpha }({\lambda }_q,t)c_{q_{\xi }+i+\nu },i=0,1,2,\dots ,k,$$

(15)

where ${\lambda }_q=1/{\mu }_q$ is a characteristic number corresponding to $e_{q_{\xi }}$. It is clear that in any case, we have $c_n\left(t\right)\to c_n,t\to 0$ (it can be established by direct calculations). In accordance with the definition given in [3] p. 17, we will say that series (14) converges to the element f in the sense $(B,\lambda ,\alpha )$ if there exists a sequence of the natural numbers ${\left\{N_j\right\}}_1^{\infty }$ such that

$$f=\underset{t\to +0}{lim}\underset{j\to \infty }{lim}\sum _{n=1}^{N_j}{c}_n\left(t\right)e_n.$$

Note that sums of the latter relation form a subsequence of the partial sums of the series (14).

To establish the main results, we need the following lemmas by Lidskii. Note that in spite of the fact that we have rewritten the lemmas in the refined form, the proof has not been changed and can be found in the paper [3]. Furthermore, considering an arbitrary compact operator $B:\mathfrak{H}\to \mathfrak{H}$ such that $\mathsf{\Theta }\left(B\right)\subset {\mathfrak{L}}_0\left(\theta \right),-\pi <\theta <\pi ,$ we put the following contour in correspondence to the operator

$$\gamma \left(B\right):=\left\{\lambda :\left|\lambda \right|=r>0,\left|\arg \lambda \right|\le \theta +\epsilon \right\}\cup \left\{\lambda :\left|\lambda \right|>r,\left|\arg \lambda \right|=\theta +\epsilon \right\},$$

(16)

where $\epsilon >0$ is an arbitrary small number, and the number r is chosen so that the operator ${(I-\lambda B)}^{-1}$ is regular within the corresponding closed circle. Here, we should note that the compactness property of B gives us the fact ${(I-\lambda B)}^{-1}\in \mathcal{B}\left(\mathfrak{H}\right),\lambda \in \mathbb{C}\backslash \mathrm{int}\gamma \left(B\right).$ It can be proved easily if we note that in accordance with the Corollary 3.3 [22] p. 268, we have $\mathrm{P}\left(B\right)\subset \mathbb{C}\backslash \overline{\mathsf{\Theta }\left(B\right)}.$

Lemma 6.

Assume that B is a compact operator, $\mathsf{\Theta }\left(B\right)\subset {\mathfrak{L}}_0\left(\theta \right),-\pi <\theta <\pi ,$ then on each ray ζ containing the point zero and not belonging to the sector ${\mathfrak{L}}_0\left(\theta \right)$ as well as the real axis, we have

$${\parallel (I-\lambda B)}^{-1}\parallel \le \frac{1}{sin\phi },\lambda \in \zeta ,$$

where $\phi =min\left\{\right|\arg \zeta -\theta |,|\arg \zeta +\theta \left|\right\}.$

Lemma 7.

Assume that the operator B satisfies conditions of Lemma 6, $f\in \mathrm{R}\left(B\right),$ then

$$\underset{t\to +0}{lim}\underset{\gamma \left(B\right)}{\int }{e}^{-{\lambda }^{\alpha }t}B{(I-\lambda B)}^{-1}fd\lambda =f,\alpha >0.$$

Lemma 8.

Assume that B is a compact operator, then, in the pole ${\lambda }_q$ of the operator ${(I-\lambda B)}^{-1},$ the residue of the vector function $e^{-{\lambda }^{\alpha }t}B{(I-\lambda B)}^{-1}f,(f\in \mathfrak{H}),\alpha >0$ equals to

$$-\sum _{\xi =1}^{m\left(q\right)}\sum _{i=0}^{k\left(q_{\xi }\right)}{e}_{q_{\xi }+i}{c}_{q_{\xi }+i}\left(t\right),$$

where $m\left(q\right)$ is a geometrical multiplicity of the q-th eigenvalue, $k\left(q_{\xi }\right)+1$ is a number of elements in the $q_{\xi }$-th Jourdan chain, the coefficients $c_{q_{\xi }+i}\left(t\right)$ are defined in accordance with formula (15).

3. Main Results

In this section, considering the class ${\stackrel{\sim }{\mathfrak{S}}}_{\alpha }$ under additional assumptions, we improve the results obtained by Lidskii [3]. As an application, we consider differential equations in the Hilbert space. We should stress that a significant refinement takes place in comparison with the reasonings [3]. We consider the operator classes under the point of view made in the latter section. Firstly, we consider a general statement with the made refinement related to the involved notion of the convergence exponent. Secondly, having formulated conditions in terms of the operator order, we produce an example establishing the fact in accordance with which the contours may be chosen in a concrete way, under the assumption $\rho =\alpha ,$ which provides a peculiar validity of the statement. Finally, we consider applications to the differential equations in the Hilbert space. For convenience, we will use auxiliary denotations

$$J=\sum _{\nu =0}^{\infty }{J}_{\nu };J^{+}=\sum _{\nu =0}^{\infty }{J}_{\nu }^{+};J^{-}=\sum _{\nu =0}^{\infty }{J}_{\nu }^{-}.$$

The structure of the proof of the following theorem completely belongs to Lidskii. However, we produce the proof, since we make a refinement corresponding to consideration of the case when a convergence exponent does not equal to the index of the Schatten–von Neumann class.

Theorem 2.

Assume that B is a compact operator, $\mathsf{\Theta }\left(B\right)\subset {\mathfrak{L}}_0\left(\theta \right),\theta <min\{\pi /2\alpha ,\pi \},B\in {\stackrel{\sim }{\mathfrak{S}}}_{\rho },0<\rho \le \alpha .$ Moreover, in the case $B\in {\stackrel{\sim }{\mathfrak{S}}}_{\rho }\backslash {\mathfrak{S}}_{\rho }$, the additional condition holds

$$\frac{n_{B^{m+1}}\left(r^{m+1}\right)}{r^{\rho }}\to 0,m=\left[\rho \right].$$

(17)

Then, a sequence of natural numbers ${\left\{N_{\nu }\right\}}_0^{\infty }$ can be chosen so that

$$\frac{1}{2\pi i}\underset{\gamma \left(B\right)}{\int }{e}^{-{\lambda }^{\alpha }t}B{(I-\lambda B)}^{-1}fd\lambda =\sum _{\nu =0}^{\infty }\sum _{q=N_{\nu }+1}^{N_{\nu +1}}\sum _{\xi =1}^{m\left(q\right)}\sum _{i=0}^{k\left(q_{\xi }\right)}{e}_{q_{\xi }+i}{c}_{q_{\xi }+i}\left(t\right),$$

moreover

$$\sum _{\nu =0}^{\infty }{∥\sum _{q=N_{\nu }+1}^{N_{\nu +1}}\sum _{\xi =1}^{m\left(q\right)}\sum _{i=0}^{k\left(q_{\xi }\right)}{e}_{q_{\xi }+i}{c}_{q_{\xi }+i}\left(t\right)∥}_{\mathfrak{H}}<\infty .$$

(18)

Proof. 

Consider a contour $\gamma \left(B\right).$ Having fixed $R>0,0<\delta <1,$ so that $R(1-\delta )=r,$ consider a monotonically increasing sequence ${\left\{R_{\nu }\right\}}_0^{\infty },R_{\nu }=R{(1-\delta )}^{-\nu +1}.$ Using Lemma 5, we obtain

$${\parallel (I-\lambda B)}^{-1}{\parallel }_{\mathfrak{H}}\le e^{{\gamma \left(\right|\lambda \left|\right)\left|\lambda \right|}^{\rho }}{\left|\lambda \right|}^m,m=\left[\rho \right],\left|\lambda \right|={\stackrel{\sim }{R}}_{\nu },R_{\nu }<{\stackrel{\sim }{R}}_{\nu }<R_{\nu +1},$$

where the function $\gamma \left(r\right)$ is defined in Lemma 5,

$$\beta \left(r\right)=r^{-\frac{\rho }{m+1}}\left(\underset{0}{\overset{r}{\int }}\frac{n_{B^{m+1}}\left(t\right)}{t}dt+r\underset{r}{\overset{\infty }{\int }}\frac{n_{B^{m+1}}\left(t\right)}{t^2}dt\right).$$

Denote by ${\gamma }_{\nu }$ a bound of the intersection of the ring ${\stackrel{\sim }{R}}_{\nu }<\left|\lambda \right|<{\stackrel{\sim }{R}}_{\nu +1}$ with the interior of the contour $\gamma \left(B\right),$ denote by $N_{\nu }$ a number of poles being contained in the set $\mathrm{int}\gamma \left(B\right)\cap \{\lambda :r<|\lambda |<{\stackrel{\sim }{R}}_{\nu }\}.$ In accordance with Lemma 8, we obtain

$$\frac{1}{2\pi i}\underset{{\gamma }_{\nu }}{\int }{e}^{-{\lambda }^{\alpha }t}B{(I-\lambda B)}^{-1}fd\lambda =\sum _{q=N_{\nu }+1}^{N_{\nu +1}}\sum _{\xi =1}^{m\left(q\right)}\sum _{i=0}^{k\left(q_{\xi }\right)}{e}_{q_{\xi }+i}{c}_{q_{\xi }+i}\left(t\right).$$

Let us estimate the above integral; for this purpose, split the contour ${\gamma }_{\nu }$ on terms ${\stackrel{\sim }{\gamma }}_{\nu }:=\{\lambda :|\lambda |={\stackrel{\sim }{R}}_{\nu },\left|\arg \lambda \right|<\theta +\epsilon \},{\gamma }_{{\nu }_{+}}:=\{\lambda :{\stackrel{\sim }{R}}_{\nu }<\left|\lambda \right|<{\stackrel{\sim }{R}}_{\nu +1},\arg \lambda =\theta +\epsilon \},{\gamma }_{{\nu }_{-}}:=\{\lambda :{\stackrel{\sim }{R}}_{\nu }<\left|\lambda \right|<{\stackrel{\sim }{R}}_{\nu +1},\arg \lambda =-\theta -\epsilon \}.$ In accordance with Lemma 5, we have

$$J_{\nu }:={∥\underset{{\stackrel{\sim }{\gamma }}_{\nu }}{\int }{e}^{-{\lambda }^{\alpha }t}B{(I-\lambda B)}^{-1}fd\lambda ∥}_{\mathfrak{H}}\le \underset{{\stackrel{\sim }{\gamma }}_{\nu }}{\int }{e}^{-{\lambda }^{\alpha }t}{∥B{(I-\lambda B)}^{-1}f∥}_{\mathfrak{H}}\left|d\lambda \right|$$

$$\le e^{{\gamma \left(\right|\lambda \left|\right)\left|\lambda \right|}^{\rho }}{\left|\lambda \right|}^{m+1}\underset{-\theta -\epsilon }{\overset{\theta +\epsilon }{\int }}{e}^{-t\mathrm{Re}{\lambda }^{\alpha }}d\arg \lambda ,\left|\lambda \right|={\stackrel{\sim }{R}}_{\nu }.$$

Using the theorem conditions, we obtain $\left|\arg \lambda \right|<\pi /2\alpha ,\lambda \in {\stackrel{\sim }{\gamma }}_{\nu },\nu =0,1,2,\dots .$ It follows that

$$\mathrm{Re}{\lambda }^{\alpha }\ge {\left|\lambda \right|}^{\alpha }cos\left[(\pi /2\alpha -\delta )\alpha \right]={\left|\lambda \right|}^{\alpha }sin\alpha \delta ,$$

where $\delta$ is a sufficiently small number. Thus, we obtain

$$J_{\nu }\le C{e}^{{\gamma \left(\right|\lambda \left|\right)\left|\lambda \right|}^{\rho }-t{\left|\lambda \right|}^{\alpha }sin\alpha \delta }{\left|\lambda \right|}^{m+1}=e^{{\left|\lambda \right|}^{\rho }\left[\gamma \right(\left|\lambda \right|)-{t\left|\lambda \right|}^{\alpha -\rho }sin\alpha \delta ]}{\left|\lambda \right|}^{m+1},m=\left[\rho \right],\left|\lambda \right|={\stackrel{\sim }{R}}_{\nu }.$$

Let us show that for a fixed t and a sufficiently large $\left|\lambda \right|,$ we have $\gamma \left(\right|\lambda \left|\right)-{t\left|\lambda \right|}^{\alpha -\rho }sin\alpha \delta <0.$ It follows directly from Lemma 2 in the case when $B\in {\mathfrak{S}}_{\rho }$ as well as in the case $B\in {\stackrel{\sim }{\mathfrak{S}}}_{\rho }\backslash {\mathfrak{S}}_{\rho }$, but here, we should involve the additional condition (17). Therefore, the series J converges. Using the analogous estimates, applying Lemma 6, we obtain

$$J_{\nu }^{+}:={∥\underset{{\gamma }_{{\nu }_{+}}}{\int }{e}^{-{\lambda }^{\alpha }t}B{(I-\lambda B)}^{-1}fd\lambda ∥}_{\mathfrak{H}}\le {C\parallel f\parallel }_{\mathfrak{H}}\underset{R_{\nu }}{\overset{R_{\nu +1}}{\int }}|e^{-t{\lambda }^{\alpha }}\left|\right|d\lambda |\le C{e}^{-t{R}_{\nu }^{\alpha }sin\alpha \epsilon }\underset{R_{\nu }}{\overset{R_{\nu +1}}{\int }}\left|d\lambda \right|$$

$$=C{e}^{-t{R}_{\nu }^{\alpha }sin\alpha \epsilon }\{R_{\nu +1}-R_{\nu }\};$$

$$J_{\nu }^{-}:={∥\underset{{\gamma }_{{\nu }_{-}}}{\int }{e}^{-{\lambda }^{\alpha }t}B{(I-\lambda B)}^{-1}fd\lambda ∥}_{\mathfrak{H}}\le C{e}^{-t{R}_{\nu }^{\alpha }sin\alpha \epsilon }\{R_{\nu +1}-R_{\nu }\}.$$

Therefore, the series $J^{+},J^{-}$ are convergent. Thus, we obtain relation (18), from which follows the rest of the theorem statement. □

3.1. Sequence of Power Type Contours

It is remarkable that we can choose a sequence of contours in various ways. For instance, a sequence of contours of the exponential type was considered in the paper [3]. In this section, we produce an application of the previous section results: we study a concrete operator class for which it is possible to choose a sequence of contours of the power type. At the same time, having involved an additional condition, we can spread the principal result of the section on a wider operator class. Note that using condition $\mathrm{H}2,$ it is not hard to prove that $\mathrm{Re}{(\stackrel{\sim }{W}f,f)}_{\mathfrak{H}}-k|\mathrm{Im}{(\stackrel{\sim }{W}f,f)}_{\mathfrak{H}}|\ge (C_2-k{C}_1){\parallel f\parallel }_{{\mathfrak{H}}_{+}}^2,k>0,$ from which follows a fact $\mathsf{\Theta }\left(A\right)\subset {\mathfrak{L}}_0\left(\theta \right),\theta =arctan(C_1/C_2).$ In general, the last relation gives us a range of the semi-angle $\pi /4<\theta <\pi /2;$ thus, the conditions $\mathrm{H}1,\mathrm{H}2$ are not sufficient to guarantee a value of the semi-angle less than $\pi /4.$ However, we should remark that some relevant results can be obtained in the case corresponding to sufficiently small values of the semi-angle; this gives us a motivation to consider a more specific additional assumption

$\left(\mathrm{H}3\right)|\mathrm{Im}{(\stackrel{\sim }{L}f,g)}_{\mathfrak{H}}|\le C_3{\parallel f\parallel }_{{\mathfrak{H}}_{+}}{\parallel g\parallel }_{\mathfrak{H}},,f,g\in \mathfrak{M},C_3>0.$

In this case, we have $\mathrm{Re}{(\stackrel{\sim }{W}f,f)}_{\mathfrak{H}}-k|\mathrm{Im}{(\stackrel{\sim }{W}f,f)}_{\mathfrak{H}}|\ge C_2{\parallel f\parallel }_{{\mathfrak{H}}_{+}}-k{C}_3\left\{{\epsilon \parallel f\parallel }_{{\mathfrak{H}}_{+}}^2/2+{\parallel f\parallel }_{\mathfrak{H}}^2/2\epsilon \right\}\ge$$(C_2-k\epsilon C_3){|f\parallel }_{{\mathfrak{H}}_{+}}^2/2+(C_2/C_0-k{C}_3/\epsilon ){|f\parallel }_{\mathfrak{H}}^2/2,k>0.$ Thus, choosing $\epsilon =C_2/k{C}_3,$ we obtain $\mathsf{\Theta }(\stackrel{\sim }{W})\subset {\mathfrak{L}}_{\iota }\left({\theta }_{\iota }\right),$ where $\iota =C_2/2C_0-{\left(k{C}_3\right)}^2/2C_2,{\theta }_{\iota }=arctan(1/k).$ This relation guarantees that we can choose a sufficiently small value of the semi-angle ${\theta }_{\iota }.$ We put the following contour in correspondence to an operator L satisfying the additional condition $\mathrm{H}3$

$$\mathsf{\Gamma }\left(A\right):=\mathrm{Fr}\left\{\left({\mathfrak{L}}_0({\theta }_0+\epsilon )\cap {\mathfrak{L}}_{\iota }({\theta }_{\iota }+\epsilon )\right)\backslash {\mathfrak{C}}_r\right\},\iota <0,{\mathfrak{C}}_r:=\left\{\lambda :\left|\lambda \right|<r,\left|\arg \lambda \right|\le {\theta }_0\right\},$$

where r is chosen so that the operator ${(I-\lambda A)}^{-1}$ is regular within the corresponding closed circle, $\epsilon >0$ is sufficiently small.

Lemma 9.

Assume that the condition $\mathrm{H}3$ holds, then

$${\parallel (I-\lambda A)}^{-1}{\parallel }_{\mathfrak{H}}\le C,\lambda \in \mathrm{Fr}\left\{{\mathfrak{L}}_0({\theta }_0+\epsilon )\cap {\mathfrak{L}}_{\iota }({\theta }_{\iota }+\epsilon )\right\},\iota <0,$$

where $\iota =C_2/2C_0-{\left(k{C}_3\right)}^2/2C_2,{\theta }_{\iota }=arctan(1/k),\epsilon >0$ is an arbitrary small number.

Proof. 

Firstly, we should note that in accordance with the condition $\mathrm{H}3,$ for an arbitrary large value $k,$ we have $\mathsf{\Theta }(\stackrel{\sim }{W})\subset {\mathfrak{L}}_{\iota }\left({\theta }_{\iota }\right),$ where $\iota =C_2/2C_0-{\left(k{C}_3\right)}^2/2C_2,{\theta }_{\iota }=arctan(1/k).$ Hence, $\mathsf{\Theta }(\stackrel{\sim }{W})\subset {\mathfrak{L}}_0\left({\theta }_0\right)\cap {\mathfrak{L}}_{\iota }\left({\theta }_{\iota }\right),$ where $\iota$ is arbitrarily negative. Therefore, $\mathsf{\Theta }\left(A\right)\subset {\mathfrak{L}}_0\left({\theta }_0\right)\cap {\mathfrak{L}}_{\iota }\left({\theta }_{\iota }\right),$ and it can be verified directly due to the geometrical methods. Note that by virtue of the Lemma 4 [3], we have ${\parallel (I-\lambda A)}^{-1}{\parallel }_{\mathfrak{H}}\le C,\lambda \in \mathrm{Fr}\left\{{\mathfrak{L}}_0({\theta }_0+\epsilon )\right\}.$ Thus, to obtain the desired result, it suffices to prove that ${\parallel (I-\lambda A)}^{-1}{\parallel }_{\mathfrak{H}}\le C,\lambda \in \mathrm{Fr}\left\{{\mathfrak{L}}_{\iota }({\theta }_{\iota }+\epsilon )\right\},\mathrm{Re}\lambda \ge 0.$ Note that in this case, $\lambda \in \mathrm{P}(\stackrel{\sim }{W})$ and we have a chain of reasonings $\forall f\in \mathfrak{H}:{(\stackrel{\sim }{W}-\lambda I)}^{-1}f=h\in \mathrm{D}(\stackrel{\sim }{W});$ $(\stackrel{\sim }{W}-\lambda I)h=f;{(f,h)}_{\mathfrak{H}}={(\stackrel{\sim }{W}h,h)}_{\mathfrak{H}}-\lambda {\parallel h\parallel }_{\mathfrak{H}}^2.$ Using the latter relation, we obtain ${|(f,h)}_{\mathfrak{H}}{|/\parallel h\parallel }_{\mathfrak{H}}^2=|{(\stackrel{\sim }{W}h,h)}_{\mathfrak{H}}/{\parallel h\parallel }_{\mathfrak{H}}^2-\lambda |\ge |\lambda -\iota |sin\epsilon .$ Therefore, using the Cauchy–Schwartz inequality, we obtain

$$\parallel {(\stackrel{\sim }{W}-\lambda I)}^{-1}{f\parallel }_{\mathfrak{H}}\le \frac{1}{|\lambda -\iota |sin\epsilon }·{\parallel f\parallel }_{\mathfrak{H}},f\in \mathfrak{H}.$$

Taking into account the fact ${(\stackrel{\sim }{W}-\lambda I)}^{-1}={(I-\lambda A)}^{-1}A=\{{(I-\lambda A)}^{-1}-I\}/\lambda ,$ we obtain ${\parallel (I-\lambda A)}^{-1}{\parallel }_{\mathfrak{H}}{-1\le \parallel (I-\lambda A)}^{-1}{-I\parallel }_{\mathfrak{H}}\le \left|\lambda \right|/|\lambda -\iota |sin\epsilon ,$ from which follows the desired result. □

Lemma 10.

Assume that the condition $\mathrm{H}3$ holds, $f\in \mathrm{R}\left(A\right),$ then

$$\underset{t\to +0}{lim}\underset{\mathsf{\Gamma }\left(A\right)}{\int }{e}^{-{\lambda }^{\alpha }t}A{(I-\lambda A)}^{-1}fd\lambda =f,\alpha >0.$$

Proof. 

The proof is analogous to the proof of the Lemma 5 [3], and the only difference is in the following: we should use Lemma 9 instead of Lemma 6. □

The theorems given below are formulated under the assumption that either the condition $\mathsf{\Theta }\left(A\right)\subset {\mathfrak{L}}_0\left(\theta \right),\theta <\pi /2\alpha$ or the condition $\mathrm{H}3$ holds. In accordance with such an alternative, we put in correspondence $\varpi :=\gamma \left(A\right)$ or $\varpi :=\mathsf{\Gamma }\left(A\right)$, respectively. The following theorem is similar to the result [3], but it is formulated in terms of the operator order. Although the principal clarification $\alpha =\rho$ has not been obtained, it can be interesting by virtue of the different ways of choosing a sequence of contours.

Theorem 3.

Assume that the operator $\stackrel{\sim }{W}$ satisfies the condition $\alpha >2/\mu ,\mu \in (0,1]$ and $\alpha >1,\mu \in (1,\infty ).$ Then, a sequence of natural numbers ${\left\{N_{\nu }\right\}}_0^{\infty }$ can be chosen so that

$$\frac{1}{2\pi i}\underset{\varpi }{\int }{e}^{-{\lambda }^{\alpha }t}A{(I-\lambda A)}^{-1}fd\lambda =\sum _{\nu =0}^{\infty }\sum _{q=N_{\nu }+1}^{N_{\nu +1}}\sum _{\xi =1}^{m\left(q\right)}\sum _{i=0}^{k\left(q_{\xi }\right)}{e}_{q_{\xi }+i}{c}_{q_{\xi }+i}\left(t\right),$$

where

$$\sum _{\nu =0}^{\infty }{∥\sum _{q=N_{\nu }+1}^{N_{\nu +1}}\sum _{\xi =1}^{m\left(q\right)}\sum _{i=0}^{k\left(q_{\xi }\right)}{e}_{q_{\xi }+i}{c}_{q_{\xi }+i}\left(t\right)∥}_{\mathfrak{H}}<\infty ,$$

(19)

the following relation holds for the eigenvalues

$$|{\lambda }_{N_{\nu }+k}|-|{\lambda }_{N_{\nu }+k-1}|\le C|{\lambda }_{N_{\nu }+k}{|}^{1-1/\tau },k=2,3,\dots ,N_{\nu +1}-N_{\nu },0<\tau <\mu .$$

Proof. 

In accordance with Theorem 1, we have

$$|{\lambda }_i^{-1}|=o\left(i^{-\tau }\right),i\to \infty ,0<\tau <\mu ,$$

Thus, using the fact ${\lambda }_i/i^{\tau }\ge C,$ we can prove that there exists a subsequence ${\left\{{\lambda }_{N_{\nu }}\right\}}_{\nu =0}^{\infty },$ such that

$$|{\lambda }_{N_{\nu }+1}|-|{\lambda }_{N_{\nu }}|\ge K|{\lambda }_{N_{\nu }+1}{|}^{1-1/\tau },K>0,$$

For this purpose, it suffices to establish the following implication

$$\underset{n\to \infty }{lim}({\lambda }_{n+1}-{\lambda }_n)/{\lambda }_n^{(p-1)/p}=0,⟹\underset{n\to \infty }{lim}{\lambda }_n/n^p=0,p>0$$

(see proof of Lemma 2 [1]). Now, consider

$$|{\lambda }_{N_{\nu }+1}|-|{\lambda }_{N_{\nu }}|\ge K|{\lambda }_{N_{\nu }}{|}^q,q:=1-1/\tau ,$$

and let us find ${\delta }_{\nu }$ from the condition $R_{\nu }=K|{\lambda }_{N_{\nu }}{|}^q+|{\lambda }_{N_{\nu }}|,R_{\nu }(1-{\delta }_{\nu })=\left|{\lambda }_{N_{\nu }}\right|,$ we have ${\delta }_{\nu }^{-1}=1+K^{-1}{\left|{\lambda }_{N_{\nu }}\right|}^{1-q}.$ Furthermore, we restrict our reasonings considering the case $\mu \in (0,1],$ since the reasonings corresponding to the case $\alpha >1,\mu \in (1,\infty )$ are absolutely analogous. Note that in accordance with Lemma 3 [3], the following relation holds

$$\sum _{i=1}^{\infty }{\lambda }_i^{\frac{\varrho }{(m+1)}}\left(\stackrel{\sim }{A}\right)\le \sum _{i=1}^{\infty }{\lambda }_i^{\varrho }\left(A\right)<\infty ,$$

where $\varrho$ is chosen so that $2/\mu <\varrho <\alpha ,m=\left[\varrho \right],\stackrel{\sim }{A}:={\left(A^{\ast m+1}{A}^{m+1}\right)}^{1/2}.$ It is clear that $\stackrel{\sim }{A}\in {\mathfrak{S}}_{\varrho /(m+1)}.$ Consider a function

$$\beta \left(r\right)=r^{-\frac{\varrho }{(m+1)}}\left(\underset{0}{\overset{r}{\int }}\frac{n_{A^{m+1}}\left(t\right)dt}{t}+r\underset{r}{\overset{\infty }{\int }}\frac{n_{A^{m+1}}\left(t\right)dt}{t^2}\right).$$

Here, we produce the variant of the proof corresponding to the case $\mathrm{H}3.$ The variant of the proof corresponding to the case $\mathsf{\Theta }\left(A\right)\subset {\mathfrak{L}}_0\left(\theta \right),\theta <\pi /2\alpha$ is analogous and left to the reader. Consider a contour $\mathsf{\Gamma }\left(A\right)$ that is absolutely analogously to the reasonings of Theorem 2; applying Lemma 5, we claim that there exists an arch ${\stackrel{\sim }{\gamma }}_{\nu }:=\{\lambda :|\lambda |={\stackrel{\sim }{R}}_{\nu },\left|\arg \lambda \right|<{\theta }_{\iota }+\epsilon \}$ in the ring $(1-{\delta }_{\nu })R_{\nu }<\left|\lambda \right|<R_{\nu },$ for which the following estimate holds for a sufficiently small value $\delta >0$

$$J_{\nu }={∥\underset{{\stackrel{\sim }{\gamma }}_{\nu }}{\int }{e}^{-{\lambda }^{\alpha }t}A{(I-\lambda A)}^{-1}fd\lambda ∥}_{\mathfrak{H}}\le C{e}^{{\left|\lambda \right|}^{\varrho }\left[\gamma \right(\left|\lambda \right|)-{t\left|\lambda \right|}^{\alpha -\varrho }sin\alpha \delta ]}{\left|\lambda \right|}^{m+1},\left|\lambda \right|={\stackrel{\sim }{R}}_{\nu },$$

where $\gamma \left(\right|\lambda \left|\right)={\beta \left(\right|\lambda |}^{m+1})+(2+ln\{12e/{\delta }_{\nu }\}){\beta \left(\right|2e\lambda |}^{m+1}){\left(2e\right)}^{\varrho }.$ It is clear that within the contour $\mathsf{\Gamma }\left(A\right)$ between the arches ${\stackrel{\sim }{\gamma }}_{\nu },{\stackrel{\sim }{\gamma }}_{\nu +1}$ (we denote the boundary of this domain by ${\gamma }_{\nu }$), there lie only those eigenvalues for which the following relation holds

$$|{\lambda }_{N_{\nu }+k}|-|{\lambda }_{N_{\nu }+k-1}|\le C|{\lambda }_{N_{\nu }+k}{|}^q,k=2,3,\dots ,N_{\nu +1}-N_{\nu }.$$

Using Lemma 8, we obtain a relation

$$\frac{1}{2\pi i}\underset{{\gamma }_{\nu }}{\int }{e}^{-{\lambda }^{\alpha }t}A{(I-\lambda A)}^{-1}fd\lambda =\sum _{q=N_{\nu }+1}^{N_{\nu +1}}\sum _{\xi =1}^{m\left(q\right)}\sum _{i=0}^{k\left(q_{\xi }\right)}{e}_{q_{\xi }+i}{c}_{q_{\xi }+i}\left(t\right).$$

Hence, to prove the main statement of the theorem, we should show that the series composed of the above terms converges. Here, we want to realize the idea of splitting ${\gamma }_{\nu }$ on terms. Let us prove that the series J converges. Substituting ${\delta }_{\nu }^{-1},$ we have $ln\{12e/{\delta }_{\nu }\}=ln\{12e+12e{K}^{-1}|{\lambda }_{N_{\nu }}{|}^{1-q}\}\le Cln\{|{\lambda }_{N_{\nu }}{|}^{1-q}\}.$ It is clear that to obtain the desired result, we should prove that

$$ln|{\lambda }_{N_{\nu }}{|}^{1-q}\beta \left(\right|{\lambda }_{N_{\nu }}{|}^{m+1})\to 0,\nu \to \infty .$$

Using simple reasonings based on the fact $ln|{\lambda }_{N_{\nu }}|/|{\lambda }_{N_{\nu }}{|}^{\epsilon }\to 0,\epsilon >0,\nu \to \infty ,$ applying Lemma 2, we obtain the desired result. Finally, we should consider the integrals along the contours ${\gamma }_{{\nu }_{+}}:=\{\lambda :(1-{\delta }_{\nu })R_{\nu }<\left|\lambda \right|<R_{\nu },\arg \lambda ={\theta }_s+\epsilon \},{\gamma }_{{\nu }_{-}}:=\{\lambda :(1-{\delta }_{\nu })R_{\nu }<\left|\lambda \right|<R_{\nu },\arg \lambda =-{\theta }_s-\epsilon \},$ where $s=0,\iota .$ Analogously to Theorem 2, applying Lemma 9, we obtain

$$J_{\nu }^{+}:={∥\underset{{\gamma }_{{\nu }_{+}}}{\int }{e}^{-{\lambda }^{\alpha }t}A{(I-\lambda A)}^{-1}fd\lambda ∥}_{\mathfrak{H}}\le {C\parallel f\parallel }_{\mathfrak{H}}\underset{(1-{\delta }_{\nu })R_{\nu }}{\overset{R_{\nu }}{\int }}|e^{-t{\lambda }^{\alpha }}\left|\right|d\lambda |\le$$

$$\le C{e}^{-t{(1-{\delta }_{\nu })}^{\alpha }{R}_{\nu }^{\alpha }sin\alpha \delta }\underset{(1-{\delta }_{\nu })R_{\nu }}{\overset{R_{\nu }}{\int }}\left|d\lambda \right|=C{e}^{-t{(1-{\delta }_{\nu })}^{\alpha }{R}_{\nu }^{\alpha }sin\alpha \delta }{\delta }_{\nu }{R}_{\nu };$$

$$J_{\nu }^{-}:={∥\underset{{\gamma }_{{\nu }_{+}}}{\int }{e}^{-{\lambda }^{\alpha }t}A{(I-\lambda A)}^{-1}fd\lambda ∥}_{\mathfrak{H}}\le C{e}^{-t{(1-{\delta }_{\nu })}^{\alpha }{R}_{\nu }^{\alpha }sin\alpha \delta }{\delta }_{\nu }{R}_{\nu }.$$

Therefore the series $J^{+},J^{-}$ are convergent. Thus, we obtain (19), from which follows the rest of the theorem statement. □

Recall that a sequence of contours of the exponential type was considered in the paper [3] under the imposed condition $\alpha >\rho .$ We improve this result in the following sense; we produce a sequence of contours of the power type, which gives us a solution of the problem in the case $A\in {\stackrel{\sim }{\mathfrak{S}}}_{\rho },\alpha =\rho .$

Theorem 4.

Assume that a normal operator $\stackrel{\sim }{W}$ satisfies the condition ${({ln}^{1+1/\alpha }x)}_{{\lambda }_i\left(H\right)}^{\prime }=o\left(i^{-1/\alpha }\right),$ $\alpha >1.$ Then, a sequence of the natural numbers ${\left\{N_{\nu }\right\}}_0^{\infty }$ can be chosen so that

$$\frac{1}{2\pi i}\underset{\varpi }{\int }{e}^{-{\lambda }^{\alpha }t}A{(I-\lambda A)}^{-1}fd\lambda =\sum _{\nu =0}^{\infty }\sum _{q=N_{\nu }+1}^{N_{\nu +1}}\sum _{\xi =1}^{m\left(q\right)}\sum _{i=0}^{k\left(q_{\xi }\right)}{e}_{q_{\xi }+i}{c}_{q_{\xi }+i}\left(t\right),$$

moreover

$$\sum _{\nu =0}^{\infty }{∥\sum _{q=N_{\nu }+1}^{N_{\nu +1}}\sum _{\xi =1}^{m\left(q\right)}\sum _{i=0}^{k\left(q_{\xi }\right)}{e}_{q_{\xi }+i}{c}_{q_{\xi }+i}\left(t\right)∥}_{\mathfrak{H}}<\infty ,$$

(20)

the following relation holds for the corresponding eigenvalues

$$|{\lambda }_{N_{\nu }+k}|-|{\lambda }_{N_{\nu }+k-1}|\le C|{\lambda }_{N_{\nu }+k}{|}^{1-1/\tau },k=2,3,\dots ,N_{\nu +1}-N_{\nu },0<\tau <1/\alpha .$$

Proof. 

Applying Theorem 1, we obtain

$$|{\lambda }_i^{-1}|=o\left(i^{-\tau }\right),i\to \infty ,0<\tau <1/\alpha ,$$

Thus, using the fact ${\lambda }_i/i^{\tau }\ge C,$ we can prove that that there exists a subsequence ${\left\{{\lambda }_{N_{\nu }}\right\}}_{\nu =0}^{\infty },$ such that

$$|{\lambda }_{N_{\nu }+1}|-|{\lambda }_{N_{\nu }}|\ge K|{\lambda }_{N_{\nu }+1}{|}^{1-1/\tau },K>0,$$

for this purpose, it suffices to establish the following implication

$$\underset{n\to \infty }{lim}({\lambda }_{n+1}-{\lambda }_n)/{\lambda }_n^{(p-1)/p}=0,⟹\underset{n\to \infty }{lim}{\lambda }_n/n^p=0,p>0$$

(see proof of Lemma 2 [1]). Now, consider

$$|{\lambda }_{N_{\nu }+1}|-|{\lambda }_{N_{\nu }}|\ge K|{\lambda }_{N_{\nu }}{|}^q,q:=1-1/\tau ,$$

and let us find ${\delta }_{\nu }$ from the condition $R_{\nu }=K|{\lambda }_{N_{\nu }}{|}^q+|{\lambda }_{N_{\nu }}|,R_{\nu }(1-{\delta }_{\nu })=\left|{\lambda }_{N_{\nu }}\right|;$ then, ${\delta }_{\nu }^{-1}=1+K^{-1}{\left|{\lambda }_{N_{\nu }}\right|}^{1-q}.$ In accordance with Lemma 4, we have $A\in {\stackrel{\sim }{\mathfrak{S}}}_{\rho },\rho \in [0,\alpha ],n_A\left(r\right)=o(r^{\alpha }/lnr).$ Here, we produce the variant of the proof corresponding to the case $\mathrm{H}3.$ The variant of the proof corresponding to the case $\mathsf{\Theta }\left(A\right)\subset {\mathfrak{L}}_0\left(\theta \right),\theta <\pi /2\alpha$ is absolutely analogous and left to the reader. Consider a contour $\mathsf{\Gamma }\left(A\right),$ applying Lemma 5 analogously to the reasonings of Theorem 2, we claim that for a sufficiently small $\delta >0,$ there exists an arch ${\stackrel{\sim }{\gamma }}_{\nu }:=\{\lambda :|\lambda |={\stackrel{\sim }{R}}_{\nu },\left|\arg \lambda \right|<\theta +\epsilon \}$ in the ring $(1-{\delta }_{\nu })R_{\nu }<\left|\lambda \right|<R_{\nu },$ on which the following estimate holds

$$J_{\nu }={∥\underset{{\stackrel{\sim }{\gamma }}_{\nu }}{\int }{e}^{-{\lambda }^{\alpha }t}A{(I-\lambda A)}^{-1}fd\lambda ∥}_{\mathfrak{H}}\le e^{{\left|\lambda \right|}^{\alpha }\left[\gamma \right(\left|\lambda \right|)-tsin\alpha \delta ]}{\left|\lambda \right|}^{m+1},m=\left[\alpha \right],\left|\lambda \right|={\stackrel{\sim }{R}}_{\nu },$$

(21)

where $\gamma \left(\right|\lambda \left|\right)={\beta \left(\right|\lambda |}^{m+1})+(2+ln\{12e/{\delta }_{\nu }\}){\beta \left(\right|2e\lambda |}^{m+1}){\left(2e\right)}^{\alpha },$

$$\beta \left(r\right)=r^{-\frac{\alpha }{m+1}}\left(\underset{0}{\overset{r}{\int }}\frac{n_{A^{m+1}}\left(t\right)dt}{t}+r\underset{r}{\overset{\infty }{\int }}\frac{n_{A^{m+1}}\left(t\right)dt}{t^2}\right).$$

It is clear that within the contour $\mathsf{\Gamma }\left(A\right),$ between the arches ${\stackrel{\sim }{\gamma }}_{\nu },{\stackrel{\sim }{\gamma }}_{\nu +1}$ (we denote the boundary of this domain by ${\gamma }_{\nu }$), there lie the eigenvalues only for which the following relation holds

$$|{\lambda }_{N_{\nu }+k}|-|{\lambda }_{N_{\nu }+k-1}|\le C|{\lambda }_{N_{\nu }+k}{|}^q,k=2,3,\dots ,N_{\nu +1}-N_{\nu }.$$

Using Lemma 8, we obtain a relation

$$\frac{1}{2\pi i}\underset{{\gamma }_{\nu }}{\int }{e}^{-{\lambda }^{\alpha }t}A{(I-\lambda A)}^{-1}fd\lambda =\sum _{q=N_{\nu }+1}^{N_{\nu +1}}\sum _{\xi =1}^{m\left(q\right)}\sum _{i=0}^{k\left(q_{\xi }\right)}{e}_{q_{\xi }+i}{c}_{q_{\xi }+i}\left(t\right).$$

It is clear that to obtain the desired result, we should prove that the series composed of the above terms converges. However, we can prove that the series J converges, which is a stronger condition. Here, we want to realize the idea of splitting ${\gamma }_{\nu }$ on terms. Consider the right-hand side of formula (21). Substituting ${\delta }^{-1},$ we have

$$ln\{12e/\delta \}=ln\{12e+12e{K}^{-1}|{\lambda }_{N_{\nu }}{|}^{1-q}\}\le Cln\{|{\lambda }_{N_{\nu }}{|}^{1-q}\}.$$

Hence, to obtain the desired result, we should prove that $ln|{\lambda }_{N_{\nu }}{|}^{1-q}\beta \left(\right|{\lambda }_{N_{\nu }}{|}^{m+1})\to 0,\nu \to \infty .$ In its own turn, using Lemma 3, we can prove the latter relation if we show that

$$lnr\frac{n_{A^{m+1}}\left(r^{m+1}\right)}{r^{\alpha }}\to 0,r\to \infty .$$

(22)

We need to establish some facts. Notice that the following operators have the same eigenvectors, i.e.,

$${\left(A^{\ast }A\right)}^{1/2}{f}_n={\mu }_n{f}_n⟺{\left(A^{s\ast }{A}^s\right)}^{1/2}{f}_n={\mu }_n^s{f}_n,s=m+1.$$

(23)

To prove this fact, firstly, let us show that $A^{\ast s}=A^{s\ast };$ then, it follows easily from the inclusion $A^{\ast s}\subset A^{s\ast }$ and the fact $\mathrm{D}\left(A^{\ast s}\right)=\mathfrak{H}.$ Thus, for a normal operator, we have ${\left(A^{\ast }A\right)}^s=A^{\ast s}{A}^s.$ Let us involve the spectral theorem for the selfadjoint non-negative operator, which is in accordance with the definition (see [31] Chapter 3), we have

$${\left(A^{\ast }A\right)}^{\vartheta }=\underset{0}{\overset{\parallel A^{\ast }A\parallel }{\int }}{\lambda }^{\vartheta }d{P}_{\lambda },\vartheta >0,$$

where the integral is understood in the Riemann sense as a limit of the partial sums, i.e.,

$$\sum _{i=0}^n{\xi }_i^{\vartheta }{P}_{\Delta {\lambda }_i}\stackrel{\mathfrak{H}}{⟶}\underset{0}{\overset{\parallel A^{\ast }A\parallel }{\int }}{\lambda }^{\vartheta }d{P}_{\lambda },\omega \to 0,$$

where $(0={\lambda }_0<{\lambda }_1<\dots <{\lambda }_n=\parallel A^{\ast }A\parallel )$ is an arbitrary splitting of the segment $[0,\parallel A^{\ast }A\parallel ],\omega :=\underset{i}{max}({\lambda }_{i+1}-{\lambda }_i),{\xi }_i$ is an arbitrary point belonging to $[{\lambda }_i,{\lambda }_{i+1}],$ the operators $P_{\Delta {\lambda }_i}$ are projectors corresponding to the selfadjoint operator. It follows easily from the well-known facts that if an additional $A^{\ast }A$ is a compact operator, then the above formula reduces to

$${\left(A^{\ast }A\right)}^{\vartheta }f=\sum _{n=1}^{\infty }{\lambda }_n^{\vartheta }{(f,{\phi }_n)}_{\mathfrak{H}}{\phi }_n,$$

where ${\left\{{\phi }_n\right\}}_1^{\infty },{\left\{{\lambda }_n\right\}}_1^{\infty }$ are sets of the eigenvectors and the eigenvalues of the operator $A^{\ast }A$, respectively. Taking into account the latter representation, an obvious fact that ${\left(A^{\ast }A\right)}^{\vartheta }$ is selfadjoint, it is not hard to prove that

$${\left(A^{\ast }A\right)}^{\frac{1}{2}·s}={\left(A^{\ast }A\right)}^{s·\frac{1}{2}}.$$

Thus, using the property $A^{\ast }A=A{A}^{\ast },$ we obtain

$${\left(A^{\ast }A\right)}^{\frac{1}{2}·s}={\left(A^{s\ast }{A}^s\right)}^{\frac{1}{2}},$$

from what follows the implication from the left-hand side of Formula (23). To obtain the contrary implication, we should establish the fact that the operator and its positive powers have the same eigenvectors. For this purpose, let us notice that

$$T^{\vartheta }{\phi }_i={\lambda }_i^{\vartheta }{\phi }_i,i\in \mathbb{N},$$

where $T:=A^{\ast }A.$ It follows that

$$T^{\vartheta }f=\sum _{n=1}^{\infty }{\lambda }_n^{\vartheta }{(f,{\phi }_n)}_{\mathfrak{H}}{\phi }_n=\sum _{n=1}^{\infty }{(f,T^{\vartheta }{\phi }_n)}_{\mathfrak{H}}{\phi }_n=\sum _{n=1}^{\infty }{(T^{\vartheta }f,{\phi }_n)}_{\mathfrak{H}}{\phi }_n,$$

therefore

$$g=\sum _{n=1}^{\infty }{(g,{\phi }_n)}_{\mathfrak{H}}{\phi }_n,g\in \mathrm{R}\left(T^{\vartheta }\right).$$

Let us assume that there exists an eigenvector h of the operator $T^{\vartheta }$ that differs from ${\phi }_i,i\in \mathbb{N}.$ Using the proved above fact, we obtain

$$T^{\vartheta }h=\sum _{n=1}^{\infty }{\lambda }_n^{\vartheta }{(h,{\phi }_n)}_{\mathfrak{H}}{\phi }_n=\zeta \sum _{n=1}^{\infty }{(h,{\phi }_n)}_{\mathfrak{H}}{\phi }_n,$$

where $\zeta$ is a corresponding eigenvalue. Multiplying (in the sense of the inner product) both sides of the latter relation on ${\phi }_k,{\phi }_{k+1},$ we obtain ${\lambda }_k^{\vartheta }=\zeta ={\lambda }_{k+1}^{\vartheta };$ this contradiction proves the desired result. Thus, we complete the proof of formula (23). To complete the proof of the relation (22), we need to mention the fact that $n_A\left(\lambda \right)=n_{A^m}\left({\lambda }^m\right)$ follows easily from the relation (23). Thus, making a substitution and using the theorem conditions, we claim that the relation (22) holds; hence, the series J converges. To complete the proof, we should note that the integrals converge uniformly along the following contours ${\gamma }_{{\nu }_{+}}:=\{\lambda :(1-{\delta }_{\nu })R_{\nu }<\left|\lambda \right|<R_{\nu },\arg \lambda ={\theta }_s+\epsilon \},{\gamma }_{{\nu }_{-}}:=\{\lambda :(1-{\delta }_{\nu })R_{\nu }<\left|\lambda \right|<R_{\nu },\arg \lambda =-{\theta }_s-\epsilon \},$ where $s=0,\iota .$ Analogously to the reasonings of Theorem 2, applying Lemma 9, we obtain

$$J_{\nu }^{+}:={∥\underset{{\gamma }_{{\nu }_{+}}}{\int }{e}^{-{\lambda }^{\alpha }t}A{(I-\lambda A)}^{-1}fd\lambda ∥}_{\mathfrak{H}}\le {C\parallel f\parallel }_{\mathfrak{H}}\underset{(1-{\delta }_{\nu })R_{\nu }}{\overset{R_{\nu }}{\int }}|e^{-t{\lambda }^{\alpha }}\left|\right|d\lambda |\le C{e}^{-t{(1-\delta )}^{\alpha }{R}_{\nu }^{\alpha }sin\alpha \delta }\underset{(1-{\delta }_{\nu })R_{\nu }}{\overset{R_{\nu }}{\int }}\left|d\lambda \right|$$

$$=C{e}^{-t{(1-{\delta }_{\nu })}^{\alpha }{R}_{\nu }^{\alpha }sin\alpha \delta }{\delta }_{\nu }{R}_{\nu };$$

$$J_{\nu }^{-}:={∥\underset{{\gamma }_{{\nu }_{-}}}{\int }{e}^{-{\lambda }^{\alpha }t}A{(I-\lambda A)}^{-1}fd\lambda ∥}_{\mathfrak{H}}\le C{e}^{-t{(1-{\delta }_{\nu })}^{\alpha }{R}_{\nu }^{\alpha }sin\alpha \delta }{\delta }_{\nu }{R}_{\nu }.$$

Therefore, the series $J^{+},J^{-}$ are convergent. Thus, we obtain relation (20), from what follows the rest of the theorem statement. □

Corollary 1.

Under the Theorem 4 assumptions, we obtain

$$f=\underset{t\to +0}{lim}\sum _{\nu =0}^{\infty }\sum _{q=N_{\nu }+1}^{N_{\nu +1}}\sum _{\xi =1}^{m\left(q\right)}\sum _{i=0}^{k\left(q_{\xi }\right)}{e}_{q_{\xi }+i}{c}_{q_{\xi }+i}\left(t\right),f\in \mathrm{D}(\stackrel{\sim }{W}).$$

This fact follows immediately from Lemmas 7 and 10, respectively.

3.2. Differential Equations in the Hilbert Space

Furthermore, we will consider that a Hilbert space $\mathfrak{H}$ consists of element-functions $u:{\mathbb{R}}_{+}\to \mathfrak{H},u:=u\left(t\right),t\ge 0$, and we will assume that if u belongs to $\mathfrak{H}$, then the fact holds for all values of the variable $t.$ Notice that under such an assumption, all standard topological properties as completeness, compactness, etc. remain correctly defined. We understand such operations as differentiation and integration in the generalized sense that is caused by the topology of the Hilbert space $\mathfrak{H}.$ The derivative is understood as the following limit

$$\frac{u(t+\Delta t)-u\left(t\right)}{\Delta t}\stackrel{\mathfrak{H}}{⟶}\frac{du}{dt},\Delta t\to 0.$$

Let $t\in \mathsf{\Omega }:=[a,b],0<a<b<\infty .$ The following integral is understood in the Riemann sense as a limit of partial sums

$$\sum _{i=0}^{n}u\left({\xi }_i\right)\Delta t_i\stackrel{\mathfrak{H}}{⟶}\underset{\mathsf{\Omega }}{\int }u\left(t\right)dt,\lambda \to 0,$$

where $(a=t_0<t_1<\dots <t_n=b)$ is an arbitrary splitting of the segment $\mathsf{\Omega },\lambda :=\underset{i}{max}(t_{i+1}-t_i),and{\xi }_i$ is an arbitrary point belonging to $[t_i,t_{i+1}].$

The sufficient condition of the last integral existence is a continuous property (see [31] p. 248), i.e., $u\left(t\right)\stackrel{\mathfrak{H}}{⟶}u\left(t_0\right),t\to t_0,\forall t_0\in \mathsf{\Omega }.$ The improper integral is understood as a limit

$$\underset{a}{\overset{b}{\int }}u\left(t\right)dt\stackrel{\mathfrak{H}}{⟶}\underset{a}{\overset{c}{\int }}u\left(t\right)dt,b\to c,c\in [0,\infty ].$$

Combining these operations, we can consider a fractional differential operator in the Riemann–Liouvile sense (see [25]), i.e., in the formal form, we have

$${\mathfrak{D}}_{-}^{1/\alpha }f\left(t\right):=-\frac{1}{\mathsf{\Gamma }(1-1/\alpha )}\frac{d}{dt}\underset{0}{\overset{\infty }{\int }}f(t+x)x^{-1/\alpha }dx,\alpha >1.$$

Let us study a Cauchy problem

$${\mathfrak{D}}_{-}^{1/\alpha }u=\stackrel{\sim }{W}u,u\left(0\right)=h\in \mathrm{D}(\stackrel{\sim }{W}),$$

(24)

in the case when the operator composition ${\mathfrak{D}}_{-}^{1-1/\alpha }\stackrel{\sim }{W}$ is accretive, we assume that $h\in \mathfrak{H}.$

Theorem 5.

Assume that the Theorem 4 conditions hold; then, there exists a solution of the Cauchy problem (24) in the form

$$u\left(t\right)=\frac{1}{2\pi i}\underset{\varpi }{\int }{e}^{-{\lambda }^{\alpha }t}A{(I-\lambda A)}^{-1}hd\lambda =\sum _{\nu =0}^{\infty }\sum _{q=N_{\nu }+1}^{N_{\nu +1}}\sum _{\xi =1}^{m\left(q\right)}\sum _{i=0}^{k\left(q_{\xi }\right)}{e}_{q_{\xi }+i}{c}_{q_{\xi }+i}\left(t\right),$$

(25)

where

$$\sum _{\nu =0}^{\infty }∥\sum _{q=N_{\nu }+1}^{N_{\nu +1}}\sum _{\xi =1}^{m\left(q\right)}\sum _{i=0}^{k\left(q_{\xi }\right)}{e}_{q_{\xi }+i}{c}_{q_{\xi }+i}\left(t\right)∥<\infty ,$$

a sequence of natural numbers ${\left\{N_{\nu }\right\}}_0^{\infty }$ can be chosen in accordance with the claim of Theorem 4. Moreover, the existing solution is unique if the operator composition ${\mathfrak{D}}_{-}^{1-1/\alpha }\stackrel{\sim }{W}$ is accretive.

Proof. 

Let us find a solution in the form (25) satisfying the initial condition (24). Below, we produce the variant of the proof corresponding to the case $\mathsf{\Theta }\left(A\right)\subset {\mathfrak{L}}_0\left(\theta \right),\theta <\pi /2\alpha .$ The variant of the proof corresponding to the case $\mathrm{H}3$ is absolutely analogous and left to the reader. Consider a contour $\gamma \left(A\right).$ Using Lemma 6, it is not hard to prove that the following integral exists

$$\frac{1}{2\pi i}\underset{\gamma \left(A\right)}{\int }{e}^{-{\lambda }^{\alpha }t}{(I-\lambda A)}^{-1}hd\lambda \in \mathfrak{H};\frac{du}{dt}=-\frac{1}{2\pi i}\underset{\gamma \left(A\right)}{\int }{e}^{-{\lambda }^{\alpha }t}{\lambda }^{\alpha }A{(I-\lambda A)}^{-1}hd\lambda \in \mathfrak{H}.$$

Note that the first relation gives us the fact $u\left(t\right)\in \mathrm{D}(\stackrel{\sim }{W}).$ Using Lemmas 5 and 6 analogously to the methods of the ordinary calculus, we can establish the following formulas

$$\underset{0}{\overset{\infty }{\int }}{x}^{-1/\alpha }dx\underset{\gamma \left(A\right)}{\int }{e}^{-{\lambda }^{\alpha }(t+x)}A{(I-\lambda A)}^{-1}hd\lambda =\underset{\gamma \left(A\right)}{\int }{e}^{-{\lambda }^{\alpha }t}A{(I-\lambda A)}^{-1}hd\lambda \underset{0}{\overset{\infty }{\int }}{x}^{-1/\alpha }{e}^{-{\lambda }^{\alpha }x}dx;$$

$$\frac{d}{dt}\underset{\gamma \left(A\right)}{\int }{\lambda }^{1-\alpha }{e}^{-{\lambda }^{\alpha }t}A{(I-\lambda A)}^{-1}hd\lambda =-\underset{\gamma \left(A\right)}{\int }\lambda e^{-{\lambda }^{\alpha }t}A{(I-\lambda A)}^{-1}hd\lambda .$$

Therefore, combining these formulas, taking into account a relation

$$\underset{0}{\overset{\infty }{\int }}{x}^{-1/\alpha }{e}^{-{\lambda }^{\alpha }x}dx=\mathsf{\Gamma }(1-1/\alpha ){\lambda }^{1-\alpha },$$

we obtain

$${\mathfrak{D}}_{-}^{1/\alpha }u=\frac{1}{2\pi i}\underset{\gamma \left(A\right)}{\int }{e}^{-{\lambda }^{\alpha }t}\lambda A{(I-\lambda A)}^{-1}hd\lambda .$$

Making a substitution using the formula $\lambda A{(I-\lambda A)}^{-1}={(I-\lambda A)}^{-1}-I,$ we obtain

$${\mathfrak{D}}_{-}^{1/\alpha }u=\frac{1}{2\pi i}\underset{\gamma \left(A\right)}{\int }{e}^{-{\lambda }^{\alpha }t}{(I-\lambda A)}^{-1}hd\lambda -\frac{1}{2\pi i}\underset{\gamma \left(A\right)}{\int }{e}^{-{\lambda }^{\alpha }t}hd\lambda =I_1+I_2.$$

The second integral is equal to zero by virtue of the fact that the function under the integral is analytical inside the intersection of the domain $\mathrm{int}\gamma \left(A\right)$ with the circle of an arbitrary radius R, and it decreases sufficiently fast on the arch of the radius $R,$ when $R\to \infty .$ Now, if we consider the expression for $u,$ we obtain the fact that u is a solution of the equation, i.e., ${\mathfrak{D}}_{-}^{1/\alpha }u=\stackrel{\sim }{W}u.$ The decomposition on the series of the root vectors (25) is obtained due to Theorem 4. Let us show that the initial condition holds in the sense $u\left(t\right)\underset{}{\overset{\mathfrak{H}}{\to }}h,t\to +0.$ It becomes clear that in the case $h\in \mathrm{D}(\stackrel{\sim }{W}),$ it suffices to apply Lemma 7, which gives us the desired result. Consider a case when h is an arbitrary element of the Hilbert space $\mathfrak{H}.$ Let us involve the accretive property of the operator composition ${\mathfrak{D}}_{-}^{1-1/\alpha }\stackrel{\sim }{W}.$ It follows from Lemma 6 that for a fixed t, the following operator is bounded

$$S_{t}h=\frac{1}{2\pi i}\underset{\gamma \left(A\right)}{\int }{e}^{-{\lambda }^{\alpha }t}A{(I-\lambda A)}^{-1}hd\lambda .$$

Let us show that $\parallel S_t\parallel \le 1,t>0.$ Firstly, assume that $h\in \mathrm{D}(\stackrel{\sim }{W});$ then, in accordance with the above, we obtain $u\left(t\right)\underset{}{\overset{\mathfrak{H}}{\to }}h,t\to +0.$ Thus, we can claim that $u\left(t\right)$ is continuous at the right-hand side of the point zero. Let us apply the operator ${\mathfrak{D}}_{-}^{1-1/\alpha }$ to both sides of relation (24). Taking into account a relation ${\mathfrak{D}}_{-}^{1-1/\alpha }{\mathfrak{D}}_{-}^{1/\alpha }u=-u^{\prime },$ we obtain $u^{\prime }+{\mathfrak{D}}_{-}^{1-1/\alpha }\stackrel{\sim }{W}u=0.$ Let us multiply both sides of the latter relation on u in the sense of the inner product; then, we obtain ${\left(u_t^{\prime },u\right)}_{\mathfrak{H}}+{({\mathfrak{D}}_{-}^{1-1/\alpha }\stackrel{\sim }{W}u,u)}_{\mathfrak{H}}=0.$ Consider a real part of the latter relation, we have $\mathrm{Re}{\left(u_t^{\prime },u\right)}_{\mathfrak{H}}+\mathrm{Re}{({\mathfrak{D}}_{-}^{1-1/\alpha }\stackrel{\sim }{W}u,u)}_{\mathfrak{H}}={\left(u_t^{\prime },u\right)}_{\mathfrak{H}}/2+{\left(u,u_t^{\prime }\right)}_{\mathfrak{H}}/2+\mathrm{Re}{({\mathfrak{D}}_{-}^{1-1/\alpha }\stackrel{\sim }{W}u,u)}_{\mathfrak{H}}.$ Therefore, ${\left({\parallel u\left(t\right)\parallel }_{\mathfrak{H}}^2\right)}_t^{\prime }=-2\mathrm{Re}{({\mathfrak{D}}_{-}^{1-1/\alpha }\stackrel{\sim }{W}u,u)}_{\mathfrak{H}}\le 0.$ Integrating both sides, we obtain

$${\parallel u\left(\tau \right)\parallel }_{\mathfrak{H}}^2-{\parallel u\left(0\right)\parallel }_{\mathfrak{H}}^2=\underset{0}{\overset{\tau }{\int }}\frac{d}{dt}{\parallel u\left(t\right)\parallel }_{\mathfrak{H}}^{2}dt\le 0.$$

The last relation can be rewritten in the form $\parallel S_t{h\parallel }_{\mathfrak{H}}\le {\parallel h\parallel }_{\mathfrak{H}},h\in \mathrm{D}(\stackrel{\sim }{W}).$ Since $\mathrm{D}(\stackrel{\sim }{W})$ is a dense set in $\mathfrak{H},$ then we obtain the desired result, i.e., $\parallel S_t\parallel \le 1.$ Now, consider the following reasonings, having assumed that $h_n\underset{}{\overset{\mathfrak{H}}{\to }}h,n\to \infty ,\left\{h_n\right\}\subset \mathrm{D}(\stackrel{\sim }{W}),h\in \mathfrak{H},$ we have ${\parallel u\left(t\right)-h\parallel }_{\mathfrak{H}}=\parallel S_t{h-h\parallel }_{\mathfrak{H}}=\parallel S_{t}h-S_t{h}_n+S_t{h}_n-h_n+h_n{-h\parallel }_{\mathfrak{H}}\le \parallel S_t\parallel ·\parallel h-h_n{\parallel }_{\mathfrak{H}}+\parallel S_t{h}_n-h_n{\parallel }_{\mathfrak{H}}+{\parallel h_n-h\parallel }_{\mathfrak{H}}.$ It is clear that if we chose n so that $\parallel h-h_n{\parallel }_{\mathfrak{H}}<\epsilon /3$ and after that chose t so that $\parallel S_t{h}_n-h_n{\parallel }_{\mathfrak{H}}<\epsilon /3,$ then we obtain ${\forall \epsilon >0,\exists \delta \left(\epsilon \right):\parallel u\left(t\right)-h\parallel }_{\mathfrak{H}}<\epsilon ,t<\delta .$ Thus, the initial condition holds. The uniqueness follows easily from the fact that ${\mathfrak{D}}_{-}^{1-1/\alpha }\stackrel{\sim }{W}$ is accretive. In this case, repeating the previous reasonings, we come to

$${\parallel g\left(\tau \right)\parallel }^2-{\parallel g\left(0\right)\parallel }_{\mathfrak{H}}^2=\underset{0}{\overset{\tau }{\int }}\frac{d}{dt}{\parallel g\left(t\right)\parallel }^{2}dt\le 0,$$

(26)

where g is a sum of two solutions $u_1$ and $u_2.$ Notice that by virtue of the initial conditions, we have $g\left(0\right)=0;$ thus, relation (26) can hold only if $g=0.$ The proof is complete. □

4. Conclusions

In this paper, we formulated the sufficient conditions of the Abel–Lidskii basis property for a sectorial non-selfadjoint operator of the special type. Having studied such an operator class, we strengthened the conditions regarding the semi-angle of the sector and weakened a great deal of conditions regarding the involved parameters. Thus, we clarified the results by Lidskii devoted to the decomposition on the root vector system of the non-selfadjoint operator. We used a technique of the entire function theory and introduced the so-called Schatten–von Neumann class of the convergence exponent. Having considered strictly accretive operators satisfying special conditions formulated in terms of the norm and used a sequence of contours of the power type, we invented a peculiar method of how to calculate a contour integral involved in the problem. Finally, we produced applications to differential equations in the abstract Hilbert space; more precisely, we studied a Cauchy problem. By virtue of the made approach, we obtained a solution analytically for the right-hand side belonging to a sufficiently wide operator class. In this regard, such operators as the Riemann–Liouville fractional differential operator, the Kipriyanov operator, the Riesz potential, and the difference operator can be involved. Moreover, we can reference to the artificially constructed normal operators for which the clarification of the Lidskii results relevantly works. Here, we should explain that we can construct a normal operator with a compact resolvent in the artificial way, having known its eigenvalues. It should be also noted that Theorem 5 covers many previously obtained results in the framework of the theory of factional differential equations. However, the main advantage is the obtained abstract formula for the solution. The norm convergence of the series representing the solution allows us to apply the methods of the approximation theory. It is also worth noticing that Theorem 5 jointly with Theorem 4 gives us the opportunity to minimize conditions imposed on the fractional order at the left-hand side of the equation, and the class of artificially constructed normal operators provides the relevance and significance of such an achievement. We hope that the invented methods and approaches create the interest and inspire the reader for further study.