On the Essential Decreasing of the Summation Order in the Abel-Lidskii Sense

Abstract

In this paper, we consider a problem of decreasing the summation order in the Abel-Lidskii sense. The problem has a significant prehistory since 1962 created by such mathematicians as Lidskii V.B., Katsnelson V.E., Matsaev V.I., Agranovich M.S. As a main result, we will show that the summation order can be decreased from the values more than a convergence exponent, in accordance with the Lidskii V.B. results, to an arbitrary small positive number. Additionally, we construct a qualitative theory of summation in the Abel-Lidkii sense and produce a number of fundamental propositions that may represent the interest themselves.

1. Introduction

1.1. Historical Review

In order to establish a harmonious connection between well-known facts, recall that the eigenvectors system of a compact selfadjoint operator forms a basis in the closure of its range. This fact can be interpreted in terms of the spectral theorem as a statement on the unit decomposition [1]. Consider a more general case corresponding to a compact non-selfadjoint operator with the numerical range of values belonging to a sector with the semi-angle less than $\pi$ and the vertex situated at the point zero [2]. Obviously, the case covers a compact non-negative selfadjoint operator. Apparently, we cannot weaken conditions upon the numerical range expecting that the basis property would be preserved, moreover the fact of the root vectors system completeness becomes non-obvious [3,4] what makes a prerequisite for a comprehensive study of the issue.

In the recent century, the problem on root vectors system completeness related to non-selfadjoint operators attracted a serious attention. The beginning of the research was laid in the paper by Keldysh M.V. 1959 [5]. In subsequent works, there were found sufficient conditions for completeness of the root vectors system 1958–1959 [6,7,8]. However, the fact is that the completeness property of the root vectors system is not sufficient for the basis property. This fact establishes a prerequisite for creation of a fundamental direction in the abstract spectral theory devoted to the basis property of the root vector system in a generalized sense. In the papers by Markus A.S. 1960 [9], 1962 [10] the problem on the series convergence in the Bari sense of subspaces was considered. In the papers 1960 [11], 1962 [2] Lidskii V.B. introduced a generalized $(A,\lambda ,s)$—method of summation for series on the root vectors based on the notion of the Abelian means considered in the monograph by Hardy G.H. 1949 [12], the parameter s is called by the order of summation. The generalization for Banach spaces was considered by Markus A.S. in 1966 [13]. In the paper by Agranovich M.S. 1976 [14] a class of non-selfadjoint elliptic operators was considered in the framework of the problem. The formula connecting spectral asymptotics corresponding to an operator with a discrete spectrum and its real component was established by Markus A.S., Matsaev V.I. in 1981 [15], the authors established the sufficient conditions for unconditional basis property in the sense of subspaces 1981 [16]. The problem on preservation of the unconditional basis property under non-selfadjoint perturbations of selfadjoint operators was considered by Motovilov A.K., Shkalikov A.A. in the paper 2019 [17]. The latest overview of the results related to the problem on decomposition on the root vectors series was represented in the paper by Shkalikov A.A. 2016 [18].

The problem on decreasing of the summation order was firstly formulated by Lidskii V.B. in the paper 1962 [19] not for the general case but for the case corresponding to the perturbation of the selfadjoint elliptic operator of the second order under the strong subordination condition. More generally, the problem was considered by Katsnelson V.E. in the Ph.D. thesis 1967 [20] (see also [21]) for perturbations of non-negative selfadjoint operators. The problem on decreasing of the summation order for operators with the spectrum belonging to the domain of the parabolic and hyperbolic type was considered by Shkalikov A.A. in 1982 [22], 1983 [23]. The detailed substantiation of a method allowing to decrease the summation order was represented by Agranovich M.S. in 1994 [24] for operators with the numerical range of values belonging to a domain of the parabolic type. Moreover, a general scale of conditions admitting convergence of the root vector series in a generalized sense such as Bari, Riesz, Abel-Lidskii senses of the series convergence was established. The clarification of the summation order was implemented in the paper by Kukushkin M.V. 2022 [25].

Apparently, the main advantage of the Lidskii V.B. [2] method is wider assumptions related to the numerical range of values comparatively with the sectorial condition let alone the operator class corresponding the numerical range of values belonging to the domain of the parabolic type [10,15,24]. Note that such a location of the numerical range of values is the inherent property of the operators with a salfadjoint senior term. At the same time, a scientific novelty and relevance appear in the very case when a senior term is not selfadjoint for there exists a comprehensive theory devoted to perturbed selfadjoint operators, see papers [3,9,10,15,17,18,26]. 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. Otherwise, the methods of the papers [27,28] become relevant since they allow us to study spectral properties of operators whether we have the mentioned above representation or not, moreover they have a natural mathematical origin that appears brightly while we are considering abstract constructions expressed in terms of the semigroup theory [29].

In this paper we consider a sectorial operator belonging to the trace class. Generally, we will show that the summation order can be decreased to the value depending on the growth of the algebraic multiplicities. In particular, we establish a remarkable fact that the summation order is an arbitrary small number for operators having a sufficiently low growth of the algebraic multiplicities. The idea to consider the growth of the algebraic multiplicities as an inherent property is principally novel in comparison with the previously obtained results and the fact that the summation order can be decreased for a sectorial operator is new. The main application of this paper results appeals to the abstract Cauchy problem for the evolution equation including the qualitative theory for fractional evolution equations [30,31]. More detailed survey related to applications to various physical-chemical processes and applications to well-known concrete evolution equations of the fractional order is given in the papers [25,28,29,32,33]. Undoubtedly, the main achievement of this paper is a constructed abstract qualitative theory creating an opportunity to solve more concrete problems let alone far reaching modifications and generalizations.

1.2. Preliminaries

Let $C,C_i,i\in {\mathbb{N}}_0$ be real positive constants. We assume that values of C can be different in formulas but values of $C_i$ are certain. Everywhere further, we consider linear densely defined operators acting in a separable complex Hilbert space $\mathfrak{H}$. Denote by $\mathcal{B}(\mathfrak{H})$ the set of linear bounded operators on $\mathfrak{H}$. Denote by $\mathrm{D}(L),\mathrm{R}(L),\mathrm{N}(L),\mathrm{P}(L)$ the domain of definition, the range, the kernel, and the resolvent set of the operator L respectively. Denote by $\Sigma (L):=\mathbb{C}\setminus \mathrm{P}(L)$ the spectrum of the operator $L.$

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

$${\parallel f\parallel }_{\mathfrak{H}}\le C{\parallel f\parallel }_{{\mathfrak{H}}_{+}},f\in {\mathfrak{H}}_{+},$$

moreover any bounded set with respect to the norm ${\mathfrak{H}}_{+}$ is compact with respect to the norm $\mathfrak{H}.$ 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 the operator L respectively. In accordance with the terminology of the monograph [34] the set $\Theta (L):=\{z\in \mathbb{C}:z={(Lf,f)}_{\mathfrak{H}},f\in \mathrm{D}(L),\parallel f{\parallel }_{\mathfrak{H}}=1\}$ is called the numerical range of the operator $L.$ Define a closed sector in the complex plain ${\mathfrak{L}}_a(\theta ):=\{z\in \mathbb{C}:|arg(z-a)|\le \theta <\pi \}\cup \{a\},$ where $a\in \mathbb{C}$ is called by the vertex and $\theta$ is called by the semi-angle of the sector. An operator L is said to be sectorial if its numerical range belongs to a sector ${\mathfrak{L}}_a(\theta ),\theta <\pi /2.$ An operator L is said to be accretive if $\mathrm{Re}{(Lf,f)}_{\mathfrak{H}}\ge 0,f\in \mathrm{D}(L),$ m-accretive if ${\{z\in \mathbb{C}:\mathrm{Re}z<0\}\subset \mathrm{P}(L),\parallel (L-\lambda I)}^{-1}{\parallel \le |\mathrm{Re}\lambda |}^{-1},\mathrm{Re}\lambda <0,$ dissipative if $\mathrm{Im}{(Lf,f)}_{\mathfrak{H}}\ge 0,f\in \mathrm{D}(L).$ An operator L is called by the operator with discrete spectrum if $0\in \mathrm{P}(L)$ and the inverse operator is compact. The dimension of the root vectors subspace corresponding to a certain eigenvalue of the operator L is called by the algebraic multiplicity of the eigenvalue. Denote by ${\mu }_j(L),j\in \mathbb{N}$ the eigenvalues of the operator $L,$ where the numbering is given in accordance with increase or decrease of their absolute value and with this numbering each eigenvalue is counted as many times as its algebraic multiplicity. Denote by $\nu ({\mu }_j)$ the algebraic multiplicity of the eigenvalue ${\mu }_j(L)$ and denote by $\nu (L)$ the sum of all algebraic multiplicities of the operator $L.$ Suppose L is a compact operator and $|L|:={(L^{\ast }L)}^{1/2},$ then the eigenvalues of the operator $|L|$ are called by the singular numbers (s-numbers) of the operator L and are denoted by $s_j(L),j=1,2,\dots ,\dim \mathrm{R}(|L|).$ If $\dim \mathrm{R}(|L|)<\infty ,$ then we put by definition $s_j=0,j>\dim \mathrm{R}(|L|).$ Assume that an operator L is compact, the following relation holds

$$\sum _{n=1}^{\infty }{s}_n^{\sigma }(L)<\infty ,0<\sigma <\infty ,$$

then L is said to be in the Schatten-von Neumann class ${\mathfrak{S}}_{\sigma }(\mathfrak{H}),$ i.e., $L\in {\mathfrak{S}}_{\sigma }(\mathfrak{H})$ in symbol. Denote by ${\mathfrak{S}}_{\infty }(\mathfrak{H})$ the set of compact operators acting in $\mathfrak{H}.$ Let L be a bounded operator acting in $\mathfrak{H},$ and assume that ${\{{\phi }_n\}}_1^{\infty },{\{{\psi }_n\}}_1^{\infty }$ a pair of orthonormal bases in $\mathfrak{H}.$ Define the absolute operator norm as follows

$${\parallel L\parallel }_2:={\left(\sum _{n,k=1}^{\infty }{|{(L{\phi }_n,{\psi }_k)}_{\mathfrak{H}}|}^2\right)}^{1/2}.$$

Consider a sequence ${\{a_j\}}_1^{\infty }\subset \mathbb{C},$ define the counting function $n(r,a_j):=\mathrm{card}\{j\in \mathbb{N}:|a_j|\le r\}.$ Assume that an operator L is compact (operator with discrete spectrum), denote by $n(r,L)$ the counting function corresponding to the sequence of the absolute values of the operator characteristic numbers (eigenvalues).

Further, we consider a compact operator $B,$ observe the sequence of its eigenvalues

$$0\ne {\mu }_1=\dots ={\mu }_{p_1}\ne {\mu }_{p_1+1}=\dots ={\mu }_{p_2}\ne {\mu }_{p_2+1}=\dots ={\mu }_{p_3}\ne \dots .$$

Analogously to the definitions accepted in the entire function theory [35], we will call the numbers $p_j,j\in \mathbb{N}$ by the principal indexes. In accordance with the above, we have

$${\Delta }_j=p_j-p_{j-1},j\in \mathbb{N},$$

where ${\Delta }_j$ denotes the algebraic multiplicity of the eigenvalue ${\mu }_{p_j},$ we formally put $p_0=0.$ Denote by ${\lambda }_j:=1/{\mu }_j,j\in \mathbb{N}$ the characteristic numbers of the operator $B.$ For the reader convenience, we use special notations for the eigenvalues and the characteristic numbers corresponding to the principal indexes $y_j:={\mu }_{p_j},z_j:={\lambda }_{p_j},j\in \mathbb{N},$ we will call them by the principal eigenvalues and the principal characteristic numbers of the operator B respectively. We also follow the definitions and notations accepted in the monographs [34,36].

2. Overview of the Supplementary Results

Following the monograph [37], we introduce some notions and facts of the entire function theory. In this subsection, we use the following notations

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

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

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

(1)

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

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

(2)

It is clear that $0\le p<\beta .$ Consider a formal infinite product

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

(3)

we will call it a canonical product and call p the genus of the canonical product. By the convergence exponent $\rho$ of the sequence ${\{a_n\}}_1^{\infty }\subset \mathbb{C},a_n\ne 0,a_n\to \infty$ we mean the greatest lower bound for such numbers $\beta$ that the series (1) converges. We need the following lemma, see Theorem 2 [38] (p. 29), Lemma 3 [37].

Lemma 1.

If the series (2) converges, then the corresponding infinite product (3) converges uniformly on every compact subset and satisfies the estimate

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

where $k_p=3e(p+1)(2+lnp),p>0,k_0=1.$

2.1. Characteristic Determinant

The well-known technique used by Lidskii V.B. [2] and others appeals to the notion of the characteristic determinant and due to this reason we produce a complete description of the object. Having chosen an orthonormal basis ${\{e_j\}}_1^{\infty }\subset \mathfrak{H}$ consider a matrix ${\{b_{ij}\}}_1^{\infty }$ of the operator $B\in {\mathfrak{S}}_{\infty },$ where

$$b_{ij}:={(B{e}_{j,}{e}_i)}_{\mathfrak{H}},i,j=1,2,\dots .$$

Assume that the finite-dimensional space ${\mathbb{E}}^n$ generated by the vectors ${\{e_j\}}_1^n$ is an invariant space of the operator $B,$ thus we have a restriction $B_n\subset B,B_n:{\mathbb{E}}^n\to {\mathbb{E}}^n.$ Denote by $det\{I-B_n\}$ the determinant of the matrix ${\{{\delta }_{ij}-b_{ij}\}}_1^n.$ It is a well-known fact that the determinant $det\{I-B_n\}$ does not depend on a basis in ${\mathbb{E}}^n$ since

$$det\{I-B_n\}=\prod _{j=1}^{\nu (B_n)}\left(1-{\mu }_j(B_n)\right),$$

where $\nu (B_n)$ is the algebraic multiplicity (dimension of the root vector subspace) of the operator $B_n.$ The latter relation shows that it is possible to make sense for the following construction

$$det\{I-B\}=\prod _{j=1}^{\nu (B)}\left(1-{\mu }_j(B)\right),B\in {\mathfrak{S}}_1,$$

where $\nu (B)\le \infty .$ The product at the right-hand side of the last relation is convergent since, we have

$$B\in {\mathfrak{S}}_1\Rightarrow \sum _{j=1}^{\nu (B)}|{\mu }_j(B)|<\infty .$$

Now, consider a formal decomposition of the determinant of the matrix with the infinite quantity of rows and columns

$$\Delta (\lambda ):=det{\{{\delta }_{ij}-\lambda b_{ij}\}}_1^{\infty }=\sum _{p=0}^{\infty }{(-1)}^p{q}_p{\lambda }^p,\lambda \in \mathbb{C},$$

where $q_0=1$ and $q_p,p=1,2,\dots$ is a sum of all central minors of the matrix ${\{b_{ij}\}}_1^{\infty }$ of the order $p,$ formed from the columns and rows with $i_1,i_2,\dots ,i_p$ numbers, i.e.,

$$q_p={\displaystyle \frac{1}{p!}}\sum _{i_1,i_2,\dots ,i_p=1}^{\infty }B\left(\begin{array}{cccc}{i}_1& i_2& \dots & i_p\\ i_1& i_2& \dots & i_p\end{array}\right).$$

Note that if $B\in {\mathfrak{S}}_1$ then in accordance with the well-known theorems (see [36]), we have

$$\sum _{n=1}^{\infty }|b_{nn}|<\infty ,\sum _{n,m=1}^{\infty }{|b_{nm}|}^2<\infty ,$$

(4)

where $b_{nm}$ is the matrix coefficients of the operator $B.$ This follows easily from the properties of the trace class operators and Hilbert-Schmidtt class operators respectively. In accordance with the von Koch H. theorem [39] conditions (4) guaranty the absolute convergence of the series $q_p.$ Moreover, the formal series $\Delta (\lambda )$ is convergent for arbitrary $\lambda \in \mathbb{C},$ therefore it represents an entire function. Analogous facts take place if we consider a formal decomposition of a minor corresponding to the matrix obtained due to deleting the l-th row and the m-th column from the initial matrix ${\{{\delta }_{ij}-\lambda b_{ij}\}}_1^{\infty }.$ Thus, we can give a meaning to the following construction

$${\Delta }^{lm}(\lambda ):=1+\sum _{p=1}^{\infty }{(-1)}^p{\lambda }^p\sum _{i_1,i_2,\dots ,i_p=1}^{\infty }B{\left(\begin{array}{cccc}{i}_1& i_2& \dots & i_p\\ i_1& i_2& \dots & i_p\end{array}\right)}_{lm},\lambda \in \mathbb{C},$$

where the used formula in brackets means a minor formed from the columns and rows with $i_1,i_2,\dots ,i_p$ numbers corresponding to the matrix obtained due to deleting the l-th row and the m-th column from the initial matrix. Conditions (4) guarantee convergence of formal series ${\Delta }^{lm}(\lambda )$ for an arbitrary $\lambda \in \mathbb{C}.$

Assume that $\lambda$ is a regular point of the operator ${(I-\lambda B)}^{-1},$ in accordance with (1.11) [2] the equation $(I-\lambda B)x=f,$ where $x,f\in \mathfrak{H},$ can be rewritten as a system in the form

$$\sum _{j=1}^{\infty }({\delta }_{ij}-\lambda b_{ij})x_j=f_i,f_i={(f,e_i)}_{\mathfrak{H}},i=1,2,\dots .$$

In accordance with [2] conditions (4) guarantee existence of the solution in the form

$${(I-\lambda B)}^{-1}f=\sum _{m=1}^{\infty }\left(\sum _{l=1}^{\infty }{(-1)}^{l+m}{\displaystyle \frac{{\Delta }^{lm}(\lambda )}{\Delta (\lambda )}}{f}_l\right)e_m,$$

(5)

where $f_l={(f,e_l)}_{\mathfrak{H}}.$ The entire function $\Delta (\lambda )$ is called by the Fredholm determinant of the operator $B.$ In accordance with the definition §4, Chapter I, [36] under assumption $B\in {\mathfrak{S}}_1$ the product $det\{I-\lambda B\}$ is called by the characteristic determinant of the operator $B.$

Since the main characteristic of the studied operators is the Schatten classification then it is rather reasonable to provide auxiliary propositions formulated in corresponding terms. The following lemma is represented in [2].

Lemma 2.

Assume that B is a compact operator, P is an arbitrary orthogonal projector in $\mathfrak{H},$ then

$$s_n(PBP)\le s_n(B),n\in \mathbb{N}.$$

The statement of the following lemma is included in the proof of Lemma 2 [2], here for the reader convenience, we represent it supplied with expended reasonings.

Lemma 3.

Assume that $B\in {\mathfrak{S}}_1,$ then the following representation holds

$$\Delta (\lambda )=\prod _{n=1}^{\infty }\left\{1-\lambda {\mu }_n(B)\right\},\lambda \in \mathbb{C},i.e.,\Delta (\lambda )=det\{I-\lambda B\},$$

the characteristic and Fredholm determinants of the operator B are coincided.

Proof. 

Firstly, we should note that in accordance with Theorem 8.1 §8, Chapter III [36], the operator B belongs to the trace class. Therefore, for an arbitrary orthonormal basis ${\{{\phi }_n\}}_1^{\infty },$ we have

$$\sum _{n=1}^{\infty }{(B{\phi }_n,{\phi }_n)}_{\mathfrak{H}}<\infty .$$

The arbitrariness in the choice of a basis gives an opportunity to claim that the series is convergent after an arbitrary transposition of the terms, from what follows that the series is absolutely convergent. Hence, the first condition (4) holds. To prove the second condition (4), we should note the inclusion ${\mathfrak{S}}_2\subset {\mathfrak{S}}_1$ and the fact that ${\mathfrak{S}}_2$ coincides with the so-called Schmidt class of operators having a finite absolute operator norm ${\parallel ·\parallel }_2.$ The latter fact can be established if we consider a complement of the orthonormal set ${\{{\phi }_n\}}_1^{\infty }$ of the eigenvectors of the operator $B^{\ast }B$ to a basis ${\{{\psi }_n\}}_1^{\infty }$ in the Hilbert space. Then in accordance with the well-known decomposition formula (see §3, Chapter V, [34]), we get the orthogonal sum

$$\mathfrak{H}=\overline{\mathrm{R}(B^{\ast }B)}\dot{+}\mathrm{N}(B^{\ast }B),$$

where ${\{{\phi }_n\}}_1^{\infty }$ is a basis in $\overline{\mathrm{R}(B^{\ast }B)},$ in accordance with the well-known property of compact selfadjoint operators. Therefore, a complement of the system ${\{{\phi }_n\}}_1^{\infty }$ to the basis in $\mathfrak{H}$ belongs to $\mathrm{N}(B^{\ast }B).$ Hence

$${\parallel B\parallel }_2^2=\sum _{n,k=1}^{\infty }|{(B{\psi }_n,{\psi }_k)}_{\mathfrak{H}}{|}^2=\sum _{n=1}^{\infty }{\parallel B{\psi }_n\parallel }_{\mathfrak{H}}^2=\sum _{n=1}^{\infty }{(B^{\ast }B{\psi }_n,{\psi }_n)}_{\mathfrak{H}}=\sum _{n=1}^{\infty }{(B^{\ast }B{\phi }_n,{\phi }_n)}_{\mathfrak{H}}=\sum _{n=1}^{\infty }{s}_n^2.$$

Therefore, by virtue of belonging to the trace class conditions (4) hold for an arbitrary chosen basis in the Hilbert space.

Now, consider an arbitrary basis ${\{{\phi }_k\}}_1^{\infty }$ and consider an orthogonal projector $P_n$ corresponding to the subspace generated by the first n basis vectors ${\phi }_1,{\phi }_2,\dots ,{\phi }_n.$ Consider a determinant

$${\Delta }^{(n)}(\lambda ):=det{\{{\delta }_{ij}-\lambda b_{ij}\}}_{ij=1}^n=\prod _{k=1}^n\left\{1-\lambda {\mu }_k(P_{n}B{P}_n)\right\}$$

Using the Weil inequalities [36], we get

$$|{\Delta }^{(n)}(\lambda )|\le \prod _{k=1}^n\left\{1+|\lambda |·|{\mu }_k(P_{n}B{P}_n)|\right\}\le \prod _{k=1}^n\left\{1+|\lambda |·|s_k(P_{n}B{P}_n)|\right\}.$$

Applying Lemma 2, we get

$$|{\Delta }^{(n)}(\lambda )|\le \prod _{k=1}^{\infty }\left\{1+|\lambda |·|s_k(B)|\right\}.$$

Passing to the limit while $n\to \infty ,$ we get

$$|\Delta (\lambda )|\le \prod _{k=1}^{\infty }\left\{1+|\lambda |·|s_k(B)|\right\}.$$

It implies, if we observe Theorem 4 (Chapter I, §4) [37] that the entire function $\Delta (\lambda )$ is of the finite order. Therefore, in accordance with Theorem 13 (Chapter I, §10) [37], it has a representation by the canonical product, i.e.,

$$\Delta (\lambda )=\prod _{n=1}^{\infty }\left\{1-\lambda {\mu }_n(B)\right\}.$$

The proof is complete. □

2.2. Abel-Lidskii Series Expansion

In accordance with the Hilbert theorem (see [36] (p. 32), [40]) the spectrum of an arbitrary compact operator B consists of the so-called normal eigenvalues, it gives us an opportunity to consider a decomposition to a direct sum of subspaces

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

(6)

corresponding to the principal eigenvalue $y_q,q\in \mathbb{N},$ both summands are invariant subspaces of the operator $B,$ the first one is a finite dimensional root subspace corresponding to the eigenvalue $y_q,\dim {\mathfrak{N}}_q=\nu (y_q)$ and the second one is a subspace wherein the operator $B-y_{q}I$ is invertible. We can choose the Jordan basis in ${\mathfrak{N}}_q$ that consists of Jordan chains of eigenvectors and root vectors of the operator $B.$ Considering the set of the principal eigenvalues and corresponding Jordan bases, we can arrange a root vectors system or following to the definition by Lidskii V.B. [2] a system of the major vectors of the operator $B.$ The Riesz integral operator is defined as follows

$${\mathcal{P}}_{q}f:=-{\displaystyle \frac{1}{2\pi i}}{\oint }_{{\Gamma }_q^{\prime }}{(B-\lambda I)}^{-1}fd\lambda ,f\in \mathfrak{H},$$

where ${\Gamma }_q^{\prime }$ is a closed contour bounding a domain containing the eigenvalue $y_q$ only. The properties of the Riesz integral operator are described in detail in §1.3, Chapter I, [36]. Consider the integral operator introduced by Lidskii V.B.

$${\mathcal{P}}_q(s,t)f=-\underset{{\Gamma }_q}{\oint }{e}^{-{\lambda }^{s}t}B{\left(I-\lambda B\right)}^{-1}fd\lambda ,f\in \mathfrak{H},s,t>0,$$

where ${\Gamma }_q$ is a closed contour bounding a domain containing the characteristic number $z_q$ only. The following fact was established by Lidskii V.B. in Lemma 5 [2].

Lemma 4.

Assume that $B\in {\mathfrak{S}}_{\infty },\Theta (B)\subset {\mathfrak{L}}_0(\theta ),\theta <\pi ,$ then

$${\displaystyle \frac{1}{2\pi i}}\underset{\vartheta }{\int }{e}^{-{\lambda }^{s}t}B{\left(I-\lambda B\right)}^{-1}fd\lambda \to f,t\to 0,f\in \mathrm{R}(B),s>0,$$

(7)

We should stress that the proof represented by Lidskii V.B. is true for an arbitrary small positive value $s.$ The proof corresponding to the case $f\in \mathfrak{H},$ under the special condition on the norm of the resolvent, is represented in [21], the idea of the proof can be found in Theorem 5.1 [18]. Below, we provide a detailed proof in terms of the operator with a discrete spectrum.

Lemma 5.

Assume that W is an operator with discrete spectrum $\Theta (W)\subset {\mathfrak{L}}_0(\theta ),\theta <\pi ,$ then the following relation holds

$$f(t):={\displaystyle \frac{1}{2\pi i}}\underset{\vartheta }{\int }{e}^{-{\lambda }^{s}t}{\left(W-\lambda I\right)}^{-1}fd\lambda \stackrel{\mathfrak{H}}{\to }f,t\to 0,f\in \mathfrak{H},0<s<\pi /2\theta .$$

Proof. 

Let us prove that

$${\parallel (W-\lambda I)}^{-1}{\parallel \le C|\lambda |}^{-1},\lambda \in \{z\in \mathbb{C}:argz=\psi \},\theta <|\psi |<\pi /2.$$

The inequality for the resolvent holds by virtue of Theorem 3.2 [34] (p. 268), since $\Theta (W)\subset {\mathfrak{L}}_0(\theta ),$ and as a result

$${\parallel (W-\lambda I)}^{-1}\parallel \le {(\mathrm{dist}\{\lambda ,\overline{\Theta (W)}\})}^{-1}\le {(\mathrm{dist}\left\{\lambda ,{\mathfrak{L}}_0(\theta )\right\})}^{-1}=\{|\lambda |sin(|\psi |-{\theta )\}}^{-1},$$

$$\lambda \in \{z\in \mathbb{C}:\arg z=\psi \}.$$

Note that $\mathrm{D}(W)$ is dense in $\mathfrak{H},$ therefore for an arbitrary element $f\in \mathfrak{H},$ we can choose a sequence ${\{f_n\}}_1^{\infty }\subset \mathrm{D}(W)$ such that

$$f_n\stackrel{\mathfrak{H}}{\to }f.$$

In accordance with (7), we have

$$f_n(t):={\displaystyle \frac{1}{2\pi i}}\underset{\vartheta }{\int }{e}^{-{\lambda }^{s}t}{(W-\lambda I)}^{-1}{f}_{n}d\lambda \stackrel{\mathfrak{H}}{\to }{f}_n,t\to 0.$$

Consider the inequality

$${\parallel f(t)-f\parallel }_{\mathfrak{H}}\le \parallel f_n(t)-f_n{\parallel }_{\mathfrak{H}}+\parallel f_n{(t)-f(t)\parallel }_{\mathfrak{H}}+{\parallel f_n-f\parallel }_{\mathfrak{H}},f\in \mathfrak{H}.$$

Thus, if we show that

$$f_n(t)\rightrightarrows f(t),n\to \infty ,$$

(8)

i.e., the sequence converges uniformly with respect to $t,$ then we obtain the desired result. Let us make a change of the variable $\lambda =\xi t^{-1/s},$ then the contour $\vartheta$ has undergone to a transformation leading to a contour ${\vartheta }^{\prime }$ with the same orientation and preserved tendency to the infinitely-distant point, we have

$$2\pi \parallel f_n{(t)-f(t)\parallel }_{\mathfrak{H}}=t^{-1/s}{∥\underset{{\vartheta }^{\prime }}{\int }{e}^{-{\xi }^s}{(W-t^{-1/s}\xi I)}^{-1}(f_n-f)d\xi ∥}_{\mathfrak{H}}\le$$

$$\le t^{-1/s}\parallel f_n{-f\parallel }_{\mathfrak{H}}\underset{{\vartheta }^{\prime }}{\int }{e}^{-\mathrm{Re}{\xi }^s}∥{(W-t^{-1/s}\xi I)}^{-1}∥|d\xi |\le C{t}^{-1/s}\parallel f_n{-f\parallel }_{\mathfrak{H}}\underset{{\vartheta }^{\prime }}{\int }{e}^{-\mathrm{Re}{\xi }^s}{t}^{1/s}{|\xi |}^{-1}|d\xi |=I_1.$$

Using the condition $0<s<\pi /2\theta ,$ we have $\mathrm{Re}{\xi }^s>C{|\xi |}^s,\xi \in {\vartheta }^{\prime },$ therefore

$$I_1\le \parallel f_n{-f\parallel }_{\mathfrak{H}}\underset{{\vartheta }^{\prime }}{\int }{e}^{-{C|\xi |}^s}{|\xi |}^{-1}|d\xi |\le C\parallel f_n{-f\parallel }_{\mathfrak{H}}.$$

The latter relation shows that (8) holds. The proof is complete. □

It is remarkable that the method for summation of the root vectors series invented by Lidskii V.B. [2] originates from the notion of the Abelian means considered in the monograph by Hardy G.H. [12] (p. 71). We can apply the original definition in the following way. Consider a formal decomposition of an element $f\in \mathfrak{H}$ on the series

$$f\sim \sum _{q=1}^{\infty }{\mathcal{P}}_{q}f.$$

(9)

The fact is that the compleatness of the root vectors system is not sufficient for the series convergence. In accordance with the definition given by Lidskii V.B. [2] series (9) is said to be summable to the element f via the method $(A,\lambda ,s)$ if the following relation holds

$$\exists {\{M_{\mu }\}}_0^{\infty }\subset \mathbb{N}:\sum _{\mu =0}^{\infty }\sum _{q=M_{\mu }+1}^{M_{\mu +1}}{\mathcal{P}}_q(s,t)f=S(t)f,$$

(10)

$$S(t)f\stackrel{\mathfrak{H}}{\to }f,t\to 0.$$

Definition 1.

Assume that

$$\exists {\{M_{\mu }\}}_0^{\infty }\subset \mathbb{N}:\sum _{\mu =0}^{\infty }\sum _{q=M_{\mu }+1}^{M_{\mu +1}}{\mathcal{P}}_q(s,t)f=S(t)f,f\in \mathfrak{M}\subset \mathfrak{H},$$

$$S(t)f\stackrel{\mathfrak{H}}{\to }f,t\to 0,$$

then the operator B is said to be in the class $\mathcal{A}(s,\mathfrak{M}),$ i.e., in symbol $B\in \mathcal{A}(s,\mathfrak{M}).$ The parameter s is called by the summation order.

3. Main Results

3.1. Splitting to the Infinite Set of the Invariant Subspaces

Denote by ${\mathfrak{M}}_k$ the closure of the linear subspace of the root vectors corresponding to an arbitrary subset of the eigenvalues ${\{{\mu }_{k_j}\}}_1^{\infty }\subset {\{{\mu }_j\}}_1^{\infty }$ of the compact operator $B.$

Lemma 6.

A compact operator B induces a compact restriction $B_k$ on the invariant subspace ${\mathfrak{M}}_k,$ moreover

$$\Sigma (B_k)={\{{\mu }_{k_j}\}}_1^{\infty }.$$

Proof. 

Let us show that the subspace ${\mathfrak{M}}_k$ is an invariant subspace of the operator $B.$ It is clear that the operator B preserves linear combinations of the root vectors since ${\mathfrak{N}}_q,q\in \mathbb{N}$ are invariant subspaces of the operator $B,$ see Formula (6). Thus, it suffices to show that the images of the limits of the root vectors linear combinations belong to ${\mathfrak{M}}_k.$ To prove the fact, consider an element g such that

$$B{f}_n\stackrel{\mathfrak{H}}{\to }g,f_n\stackrel{\mathfrak{H}}{\to }f\in {\mathfrak{M}}_k,n\to \infty ,f_n:=\sum _{\nu =0}^n{e}_{\nu }{c}_{n\nu },$$

where $c_{n\nu }$ are complex valued coefficients, $e_{\nu }$ root vectors corresponding to the set ${\{{\mu }_{k_j}\}}_1^{\infty }.$ In accordance with the above, we have $B{f}_n\in {\mathfrak{M}}_k.$ Since ${\mathfrak{M}}_k$ is a closed subspace in the sense of the norm of the Hilbert space $\mathfrak{H},$ then $g\in {\mathfrak{M}}_k.$ Therefore, ${\mathfrak{M}}_k$ is an invariant subspace of the operator $B.$ Note that the restriction $B_k$ is compact, since B is compact.

Let us prove the fact $\Sigma (B_k)={\{{\mu }_{k_j}\}}_1^{\infty }.$ Note that in accordance with the Hilbert theorem the spectrum of a compact operator except for the point zero consists of normal eigenvalues. Thus, it suffices to prove that the set of the eigenvalues of the operator $B_k$ coincides with the set ${\{{\mu }_{k_j}\}}_1^{\infty }.$ Consider the set of the eigenvalues ${\{{\mu }_{n_j}\}}_1^{\infty }={\{{\mu }_j\}}_1^{\infty }\setminus {\{{\mu }_{k_j}\}}_1^{\infty },$ then in accordance with Theorem 6.17, Chapter III [34], we have the decomposition

$$\mathfrak{H}={\mathfrak{M}}_l^{\prime }\oplus {\mathfrak{M}}_l^{″},l\in \mathbb{N},$$

corresponding to the finite set ${\{{\mu }_{n_j}\}}_1^l,$ where ${\mathfrak{M}}_l^{\prime }$ is a finite dimensional invariant subspace of the operator B generated by the root vectors corresponding to ${\{{\mu }_{n_j}\}}_1^l,$ and ${\mathfrak{M}}_l^{″}$ is its parallel complement respectively, we have $P_l\mathfrak{H}={\mathfrak{M}}_l^{\prime },(I-P_l)\mathfrak{H}={\mathfrak{M}}_l^{″},$ where $P_l$ is the corresponding projector, i.e.,

$$P_{l}f:=-{\displaystyle \frac{1}{2\pi i}}{\oint }_{{\Gamma }_l}{(B-\lambda I)}^{-1}fd\lambda ,f\in \mathfrak{H},$$

the contour ${\Gamma }_l$ is a closed contour bounding a domain containing the set of the eigenvalues ${\{{\mu }_{n_j}\}}_1^l$ only. Observe that the operator $P_l$ is bounded in the Hilbert space $\mathfrak{H}.$ It can be proved easily since the space ${\mathfrak{M}}_l^{\prime }$ is finite dimensional, thus using the orthogonalization procedure in the Hilbert space $\mathfrak{H}$ having preserved the basis vectors belonging to ${\mathfrak{M}}_l^{\prime }$ we easily obtain

$${(P_{l}f,P_{l}f)}_{\mathfrak{H}}\le {(f,f)}_{\mathfrak{H}},$$

from what follows that $\parallel P_l\parallel \le 1.$ Let us show that the subspace ${\mathfrak{M}}_l^{″}$ is closed, assume that

$$g_k\stackrel{\mathfrak{H}}{\to }g,k\to \infty ,{\{g_k\}}_1^{\infty }\subset {\mathfrak{M}}_l^{″},$$

in accordance with the continuous property of the operator $P_l,$ taking into account $P_l{g}_k=0,$ we get

$$\parallel P_l{g\parallel }_{\mathfrak{H}}=\parallel P_l(g_k-g){\parallel }_{\mathfrak{H}}\le {\parallel g_k-g\parallel }_{\mathfrak{H}},\Rightarrow P_{l}g=0,\Rightarrow g\in {\mathfrak{M}}_l^{″}.$$

Thus, we conclude that the space ${\mathfrak{M}}_l^{″}$ is closed. Note that in accordance with Theorem 6.17, Chapter III [34], we have $P_{l}e=0,$ where e is a root vector corresponding to the eigenvalue $\mu \in {\{{\mu }_j\}}_1^{\infty }\setminus {\{{\mu }_{n_j}\}}_1^l.$ Hence the closure of the root vectors linear combinations corresponding to ${\{{\mu }_j\}}_1^{\infty }\setminus {\{{\mu }_{n_j}\}}_1^l$ belongs to ${\mathfrak{M}}_l^{″}.$ Therefore ${\mathfrak{M}}_k\subset {\mathfrak{M}}_l^{″},l\in \mathbb{N},$ since ${\{{\mu }_j\}}_1^{\infty }\setminus {\{{\mu }_{n_j}\}}_1^l\supset {\{{\mu }_{k_j}\}}_1^{\infty }.$

Note that in accordance with the made assumptions the root vectors system corresponding to the set of the eigenvalues ${\{{\mu }_{k_j}\}}_1^{\infty }$ belongs to the root vectors system of the operator $B_k.$ Let us show that they are coincided, i.e., there does not exist a root vector of the operator $B_k$ corresponding to an eigenvalue that differs from ${\{{\mu }_{k_j}\}}_1^{\infty }.$ Assume the contrary, then taking into account the fact $B_k\subset B,$ we should admit that there exists a number N and an eigenvalue $\mu \in {\{{\mu }_{n_j}\}}_1^N,$ so that ${(B-\mu I)}^{\xi }e=0,e\in {\mathfrak{M}}_k,\xi \in \mathbb{N}.$ Hence ${\mathfrak{M}}_k\cap {\mathfrak{M}}_p^{\prime }\ne 0,p\ge N$ but it contradicts the proved above fact in accordance with which ${\mathfrak{M}}_k\subset {\mathfrak{M}}_p^{″},$ since ${\mathfrak{M}}_p^{\prime }\cap {\mathfrak{M}}_p^{″}=0.$ Therefore the root vectors system of the operator $B_k$ coincides with the root vectors system of the operator B corresponding to the set of the eigenvalues ${\{{\mu }_{k_j}\}}_1^{\infty }.$ It implies that the set of the eigenvalues of the operator $B_k$ coincides with the set ${\{{\mu }_{k_j}\}}_1^{\infty }.$ The proof is complete. □

3.2. Splitting of the Counting Function

Consider a subsequence of the natural numbers

$$N_{\nu }=\sum _{k=0}^{\nu }[{\nu }^{\beta }-k^{\beta }],\beta >0,\nu \in {\mathbb{N}}_0.$$

(11)

Let us split the sequence of the principal characteristic numbers ${\{z_j\}}_1^{\infty }$ on the groups ${\{z_{k_j}\}}_1^{\infty },$ i.e.,

$${\{z_j\}}_1^{\infty }=\bigcup _{k=0}^{\infty }{\{z_{k_j}\}}_1^{\infty },$$

(12)

corresponding to the numbers $N_{k\nu }:=[{\nu }^{\beta }-k^{\beta }];N_{k\nu }=0,\nu ⩽k$ so that the disk $\{z:|z|\le |z_{N_{\nu }}|\},$ where we formally put $z_0:=0,$ contains $N_{k\nu }$ elements of the k-th group ${\{z_{k_j}\}}_1^{\infty }.$ In terms of counting functions, we have

$$n(|z_{N_{\nu }}|,z_j)=N_{\nu }=\sum _{k=0}^{\nu }[{\nu }^{\beta }-k^{\beta }]=\sum _{k=0}^{\nu }n(|z_{N_{\nu }}|,z_{k_j}).$$

Here, we ought to point out that ${\{k_j\}}_1^{\infty }$ is a subsequence of natural numbers defined by the index k and in accordance with the last union, we have

$$\mathbb{N}=\bigcup _{k=0}^{\infty }{\{k_j\}}_1^{\infty }.$$

It is clear that splitting (12) induces, in the natural way, the splitting

$${\{{\lambda }_j\}}_1^{\infty }=\bigcup _{k=0}^{\infty }{\{{\lambda }_{k_j}\}}_1^{\infty }.$$

In accordance with the above, we can express the principal index

$$p_{N_{\nu }}=\sum _{k=0}^{\nu }\sum _{j=1}^{[{\nu }^{\beta }-k^{\beta }]}{\Delta }_j(k),$$

where ${\Delta }_j(k)$ denotes algebraic multiplicity corresponding to the principal characteristic number $z_{k_j}\in {\{z_{k_j}\}}_1^{\infty }.$ For a convenient form of writing, we will use the following shortages

$${\tilde{N}}_{k\nu }:=\sum _{j=1}^{[{\nu }^{\beta }-k^{\beta }]}{\Delta }_j(k),{\tilde{N}}_{\nu }:=p_{N_{\nu }}.$$

It is rather clear that the disk on the complex plane $\{z\in \mathbb{C}:|z|\le |{\lambda }_{{\tilde{N}}_{\nu }}|\}$ contains ${\tilde{N}}_{k\nu }$ characteristic numbers belonging to the k-th group ${\{{\lambda }_{k_j}\}}_1^{\infty },$ where we put ${\lambda }_0:=0$ in correspondence with the formalities accepted above.

Further, applying Lemma 6, we put the operator $B_k$ in correspondence with the k-th group ${\{{\lambda }_{k_j}\}}_1^{\infty }$ and use the following notation ${\lambda }_j(B_k):={\lambda }_{k_j}.$

Definition 2.

Assume that

$${\Delta }_j<C{j}^{\varphi },0<\varphi <1,j\in \mathbb{N},$$

then we will say that the operator B has the sequence of the algebraic multiplicities of the ϕ-th growth (of the lowest growth if ϕ can be chosen arbitrary small).

Lemma 7.

Assume that $B\in {\mathfrak{S}}_{\sigma },0<\sigma <\infty ,$ has the sequence of the multiplicities of the ϕ-th growth, then

$$\underset{r\to \infty }{lim}{\displaystyle \frac{n(r,B_k)}{r^s}}=0,s>\sigma \left({\displaystyle \frac{\beta }{\beta +1}}+\varphi \right),$$

uniformly with respect to $k\in {\mathbb{N}}_0.$

Proof. 

Consider a subsequence of the natural numbers ${\{N_{\nu }\}}_0^{\infty }$ defined in (11). Let us prove the following asymptotic formula

$$N_{\nu }\sim {\displaystyle \frac{\beta }{\gamma }}{\nu }^{\gamma },\nu \to \infty ,$$

(13)

here and further $\gamma :=\beta +1.$ For this purpose, we will estimate the given sum by a corresponding definite integral, i.e., calculating the integral, we have on the one hand

$$\sum _{k=1}^{\nu }{k}^{\beta }\le \underset{1}{\overset{\nu +1}{\int }}{x}^{\beta }dx={\displaystyle \frac{{(\nu +1)}^{\gamma }}{\gamma }}-{\displaystyle \frac{1}{\gamma }},$$

on the other hand

$$\sum _{k=1}^{\nu }{k}^{\beta }=\sum _{k=2}^{\nu }{k}^{\beta }+1\ge \underset{1}{\overset{\nu }{\int }}{x}^{\beta }dx+1={\displaystyle \frac{{\nu }^{\gamma }}{\gamma }}+{\displaystyle \frac{\beta }{\gamma }}.$$

Therefore

$$N_{\nu }=\sum _{k=0}^{\nu -1}[{\nu }^{\beta }-k^{\beta }]\ge {\nu }^{\gamma }-\nu -{\displaystyle \frac{{\nu }^{\gamma }}{\gamma }}+{\displaystyle \frac{1}{\gamma }}={\displaystyle \frac{\beta {\nu }^{\gamma }}{\gamma }}-\nu +{\displaystyle \frac{1}{\gamma }}.$$

Analogously

$$N_{\nu }=\sum _{k=0}^{\nu }[{\nu }^{\beta }-k^{\beta }]\le \sum _{k=0}^{\nu }({\nu }^{\beta }-k^{\beta })\le {\nu }^{\beta }(\nu +1)-{\displaystyle \frac{{\nu }^{\gamma }}{\gamma }}-{\displaystyle \frac{\beta }{\gamma }}={\displaystyle \frac{\beta {\nu }^{\gamma }}{\gamma }}+{\nu }^{\beta }-{\displaystyle \frac{\beta }{\gamma }},$$

from what follows the desired result. Note that in accordance with the fact that the operator belongs to the Schatten-von Neumann class ${\mathfrak{S}}_{\sigma },$ we have

$$\underset{r\to \infty }{lim}{\displaystyle \frac{n(r,B)}{r^{\sigma }}}=0.$$

This fact obviously follows from the implication

$$B\in {\mathfrak{S}}_{\sigma }\Rightarrow s_n(B)=o(n^{-1/\sigma })\Rightarrow |{\mu }_n(B)|=o(n^{-1/\sigma }),n\to \infty ,0<\sigma <\infty ,$$

(14)

see $8^{\circ },$ §7, Chapter III [36], Corollary 3.2, §3, Chapter II [36]. Note that ${\Delta }_j\ge 1,$ hence

$${\tilde{N}}_{\nu }=\sum _{k=0}^{\nu }\sum _{j=1}^{[{\nu }^{\beta }-k^{\beta }]}{\Delta }_j(k)\ge \sum _{k=0}^{\nu }[{\nu }^{\beta }-k^{\beta }]=N_{\nu }.$$

Therefore, applying asymptotic Formula (13), we obtain

$${\displaystyle \frac{{\nu }^{\gamma }}{|{\lambda }_{{\tilde{N}}_{\nu }}{|}^{\sigma }}}\le C{\displaystyle \frac{{\tilde{N}}_{\nu }}{|{\lambda }_{{\tilde{N}}_{\nu }}{|}^{\sigma }}}\le C_{\nu },C_{\nu }\to 0,\nu \to \infty .$$

(15)

Note that in accordance with the splitting, assuming that $N_{k\nu }>0,$ we have

$$z_{k_j}\in \{z\in \mathbb{C}:|z|\le z_{N_{\nu }}\},j=1,2,\dots ,N_{k\nu },$$

therefore using the condition imposed upon the growth of the algebraic multiplicities, we obtain

$${\Delta }_j(k)\le \underset{j\in [1,N_{\nu }]}{max}{\Delta }_j\le C{N}_{\nu }^{\varphi }.$$

Applying Formula (13), we get

$${\tilde{N}}_{k\nu }=\sum _{j=1}^{[{\nu }^{\beta }-k^{\beta }]}{\Delta }_j(k)\le \sum _{j=1}^{[{\nu }^{\beta }-k^{\beta }]}C{N}_{\nu }^{\varphi }=C[{\nu }^{\beta }-k^{\beta }]N_{\nu }^{\varphi }\le C{\nu }^{\beta +\varphi \gamma }.$$

(16)

Assuming that k is fixed, consider a sequence

$${\displaystyle \frac{{\tilde{N}}_{k\nu }}{|{\lambda }_{{\tilde{N}}_{k\nu }}(B_k){|}^s}},s>\sigma \left({\displaystyle \frac{\beta }{\gamma }}+\varphi \right),\nu =k+1,k+2,\dots .$$

Observe that the numbers ${\tilde{N}}_{k\nu }$ have variations only corresponding to the the values of the index $\nu$ satisfying the condition $\left[{(\nu +1)}^{\beta }-k^{\beta }\right]>[{\nu }^{\beta }-k^{\beta }],$ in this case, we have

$$|{\lambda }_{{\tilde{N}}_{\nu }}|<|{\lambda }_{{\tilde{N}}_{k\nu }+q}(B_k)|\le |{\lambda }_{{\tilde{N}}_{\nu +1}}|,0<q\le {\tilde{N}}_{k\nu +1}-{\tilde{N}}_{k\nu }.$$

Using the lower estimate, applying (16), we get

$${\displaystyle \frac{{\tilde{N}}_{k\nu }+q}{|{\lambda }_{{\tilde{N}}_{k\nu }+q}(B_k){|}^s}}<{\displaystyle \frac{{\tilde{N}}_{k\nu +1}}{|{\lambda }_{{\tilde{N}}_{\nu }}{|}^s}}\le C{\left\{{\displaystyle \frac{{\nu }^{\gamma }}{|{\lambda }_{{\tilde{N}}_{\nu }}{|}^{\sigma }}}\right\}}^{{\displaystyle \frac{\beta }{\gamma }}+\varphi }\le C{C}_{\nu }^{{\displaystyle \frac{\beta }{\gamma }}+\varphi },$$

from what follows the desired result. The proof is complete. □

Using the latter result, we can represent a scheme of reasonings allowing to decrease the summation order. The following paragraph is devoted to a sharper estimate for the canonical product, however the final aim is to improve the estimate for the norm of the resolvent what can be implemented due to the properties of the given above artificially constructed subsequence of the eigenvalues. Since the result is fundamental and relates to the issue in the framework of the infinite determinant theory Chapter IV [36], we may claim that it may represent the interest itself.

3.3. Sharper Estimate for the Canonical Product

Lemma 8.

Assume that B is a compact operator $B\in {\mathfrak{S}}_1,\Theta (B)\subset {\mathfrak{L}}_0(\theta ),\theta <\pi /4,$ then the following estimate holds

$$\prod _{n=1}^{\infty }|1-\lambda {\mu }_n(Q_{1}B{Q}_1)|\le \prod _{n=1}^{\infty }|1+\lambda {\mu }_n(B)|,|\arg \lambda |<\pi /4,$$

where $Q_1$ is the orthogonal projector corresponding to the orthogonal complement of the one-dimensional subspace generated by an element $f\in \mathfrak{H}.$

Proof. 

Firstly, we should note that in accordance with Lemma 1 [2], we have

$$s_n(Q_{1}B{Q}_1)\le s_n(B),n=1,2,\dots ,$$

hence $Q_{1}B{Q}_1\in {\mathfrak{S}}_1.$ Note that by virtue of the relation

$$\mathrm{Re}(Q_{1}B{Q}_{1}f,f)=\mathrm{Re}(B{Q}_{1}f,Q_{1}f)\ge 0,f\in \mathfrak{H},$$

we obtain the fact $\Theta (Q_{1}B{Q}_1)\subset \Theta (B).$ Consider the operators $B(\lambda )=(I+\lambda B)$ and $B_1(\lambda ):=(Q_1+\lambda B_1),B_1:=Q_{1}B{Q}_1.$ Note that

$$B^{\ast }(\lambda )B(\lambda )=I+C(\lambda ),C(\lambda )={|\lambda B|}^2+2\mathfrak{Re}(\lambda B),$$

$$B_1^{\ast }(\lambda )B_1(\lambda )=Q_1+C_1(\lambda ),C_1(\lambda ):={|\lambda B_1|}^2+2\mathfrak{Re}(\lambda B_1),$$

where ${|B|}^2:=B^{\ast }B.$ It is clear that since $C(\lambda )$ is compact selfadjoint then the set of the eigenvectors is complete in $\overline{\mathrm{R}(C(\lambda ))},$ and we can choose a basis ${\{e_k\}}_1^{\infty }$ in $\overline{\mathrm{R}(C(\lambda ))}$ such that the operator matrix will have a diagonal form—the eigenvalues are situated on the major diagonal. It is clear that the same reasonings are true for the operator $C_1(\lambda ).$ Applying Corollary 2.2, §2, Chapter II [36], we obtain easily the fact $C(\lambda ),C_1(\lambda )\in {\mathfrak{S}}_1,$ therefore, applying Corollary 1.1, $3^{\circ },$ Chapter IV, §1 [36], we get

$$\underset{n\to \infty }{lim}det\{P_n+P_{n}C(\lambda )P_n\}=det\{I+C(\lambda )\},$$

where $P_n,n\in \mathbb{N}$ is an orthogonal projector into the subspace generated by the eigenvectors ${\{e_k\}}_1^n$ of the operator $C(\lambda ).$ Note that $Q_1{P}_{1n}=P_{1n},$ where $P_{1n}$ is an orthogonal projector into the subspace generated by the eigenvectors ${\{e_{1k}\}}_1^n$ of the operator $C_1(\lambda ).$ Analogously, we get

$$\underset{n\to \infty }{lim}det\{P_{1n}+P_{1n}{C}_1(\lambda )P_{1n}\}=det\{I+C_1(\lambda )\}.$$

Consider

$$(C_1(\lambda )f,f)={|\lambda |}^2(Q_1{B}^{\ast }{Q}_{1}B{Q}_{1}f,f)+2(\mathfrak{Re}(\lambda B_1)f,f)\le {|\lambda |}^2(B^{\ast }B{Q}_{1}f,Q_{1}f)+2(\mathfrak{Re}(\lambda B)Q_{1}f,Q_{1}f)=$$

$$=(Q_{1}C(\lambda )Q_{1}f,f),$$

here we used the obvious relations

$$(Q_1{B}^{\ast }{Q}_{1}B{Q}_{1}f,f)\le (B^{\ast }B{Q}_{1}f,Q_{1}f),\mathfrak{Re}(\lambda B_1)=Q_1\mathfrak{Re}(\lambda B)Q_1.$$

Since $C(\lambda ),|arg\lambda |<\pi /4$ is a compact non-negative selfadjoint operator, then by virtue to the minimax principle for the eigenvalues (see Courant theorem [41] (p. 120)), we get

$${\mu }_n(C_1(\lambda ))\le {\mu }_n(Q_{1}C(\lambda )Q_1)\le {\mu }_n(C(\lambda )),n\in \mathbb{N},$$

therefore

$$det\{P_{1n}+P_{1n}{C}_1(\lambda )P_{1n}\}\le det\{P_n+P_{n}C(\lambda )P_n\},n\in \mathbb{N}.$$

Passing to the limit, we get

$$det\{I+C_1(\lambda )\}\le det\{I+C(\lambda )\}.$$

(17)

On the other hand, in accordance with Theorem 2.3 Chapter V [36] the system of the root vectors (including the root vectors corresponding to the zero eigenvalue) of the operator $iB$ is compleat in $\mathfrak{H}.$ Indeed, we will prove it if we show that $iB$ is dissipative and $iB\in {\mathfrak{S}}_1.$ Taking into account the fact $\Theta (B)\subset {\mathfrak{L}}_0(\theta ),\theta <\pi /4,$ we conclude that

$$\mathrm{Im}(iBf,f)=\mathrm{Re}(Bf,f)\ge 0,f\in \mathfrak{H}.$$

Therefore, the operator $iB$ is dissipative. It is clear that $s_n(iB)=s_n(B),n\in \mathbb{N},$ hence $iB\in {\mathfrak{S}}_1.$ Thus, we obtain the desired result. Note that in accordance with Theorem 2.3 Chapter V [36], we have

$$\mathfrak{H}={\mathfrak{C}}_0(iB)\dot{+}{\mathfrak{H}}_0(iB),$$

where ${\mathfrak{C}}_0(iB)$ is the invariant subspace generated by the closure of the linear combinations of the root vectors corresponding to non-real eigenvalues of the operator $iB$ and ${\mathfrak{H}}_0(iB)$ is the invariant subspace on which the restriction of $iB$ is selfadjoint. Since $\Theta (B)\subset {\mathfrak{L}}_0(\theta )$ then ${\mathfrak{C}}_0(iB)=\mathfrak{C}(iB)=\mathfrak{C}(B)$ the latter symbol denotes the closure of the linear combinations of the root vectors of the operator $B,$ we used the fact that the operators B and $iB$ have the same root vectors. It implies that ${\mathfrak{H}}_0(iB)=\mathrm{N}(B),$ since the operator $iB$ does not have real eigenvalues. Hence

$$\mathfrak{H}=\mathfrak{C}(B)\dot{+}\mathrm{N}(B).$$

Therefore, in accordance with Lemma 4.1 Chapter I [36], we can construct an orthogonal Schur basis ${\{{\omega }_j\}}_1^{\infty }\subset \mathfrak{C}$ so that the matrix of the operator induced in $\mathfrak{C}$ has a triangle form. Choosing an arbitrary basis in the space $\mathrm{N}(B),$ uniting bases of the orthogonal decomposition, we obtain the fact that the matrix of the operator B has a triangle form in a newly constructed united basis. Thus, choosing orthogonal projectors corresponding to n-dimensional subspaces, the property of the triangle determinant, we have

$$det\{P_{n}B(\lambda )P_n\}=\prod _{k=1}^n\{1+\lambda {\mu }_k(B)\};det\{P_n{B}^{\ast }(\lambda )P_n\}=det\{{[P_{n}B(\lambda )P_n]}^{\ast }\}=\prod _{k=1}^n\{1+\overline{\lambda {\mu }_k(B)}\},$$

therefore

$$det\{P_n+P_{n}C(\lambda )P_n\}=det\{P_{n}B(\lambda )P_n\}det\{P_n{B}^{\ast }(\lambda )P_n\}=\prod _{k=1}^n{|1+\lambda {\mu }_k(B)|}^2.$$

Analogously, we get

$$det\{P_{1n}+P_{1n}{C}_1(\lambda )P_{1n}\}=det\{P_{1n}{B}_1(\lambda )P_{1n}\}det\{P_{1n}{B}_1^{\ast }(\lambda )P_{1n}\}=\prod _{k=1}^n{|1+\lambda {\mu }_k(Q_{1}B{Q}_1)|}^2.$$

Applying Corollary 1.1, $2^{\circ },$ Chapter IV, §1 [36], we conclude that

$$\underset{n\to \infty }{lim}det\{P_n+P_{n}C(\lambda )P_n\}=det\{I+C(\lambda )\},\underset{n\to \infty }{lim}det\{P_{1n}+P_{1n}{C}_1(\lambda )P_{1n}\}=det\{I+C_1(\lambda )\}.$$

Taking into account (17), we get

$$\prod _{n=1}^{\infty }|1+\lambda {\mu }_n(Q_{1}B{Q}_1)|\le \prod _{n=1}^{\infty }|1+\lambda {\mu }_n(B)|.$$

Having noticed the fact

$$|1-\lambda {\mu }_n(Q_{1}B{Q}_1)|\le |1+\lambda {\mu }_n(Q_{1}B{Q}_1)|,|arg\lambda |<\pi /4,$$

we obtain the desired result. The proof is complete. □

Lemma 9.

Assume that $B\in {\mathfrak{S}}_{\sigma },0<\sigma \le 1,\Theta (B)\in {\mathfrak{L}}_0(\theta ),\theta <\pi /4,$ the following relation holds

$$\underset{r\to \infty }{lim}{\displaystyle \frac{n(r,B)}{r^s}}=0,0<c<s<\sigma .$$

Then for arbitrary numbers $R,\delta$ such that $R>0,0<\delta <1,$ there exists $(1-\delta )R<\tilde{R}<R,$ so that the following estimate holds

$${\parallel (I-\lambda B)}^{-1}\parallel \le 2e^{h\left(2eR\right)ln{\displaystyle \frac{45e^4}{\delta }}},|\lambda |=\tilde{R},|arg\lambda |<\pi /4,$$

where

$$h(r)=\left(\underset{0}{\overset{r}{\int }}{\displaystyle \frac{n(t,B)dt}{t}}+r\underset{r}{\overset{\infty }{\int }}{\displaystyle \frac{n(t,B)dt}{t^2}}\right).$$

Proof. 

Consider a Fredholm determinant of the operator $B,$ in accordance with Lemma 3 it has a representation

$$\Delta (\lambda )=\prod _{n=1}^{\infty }\left\{1-\lambda {\mu }_n(B)\right\},\lambda \in \mathbb{C}.$$

Let us chose an arbitrary element $f\in \mathfrak{H}$ and construct a new orthonormal basis having put f as a first basis element. Note that relations (4) hold for the matrix coefficients of the operator B in a new basis, this fact follows from the well-known theorem for the operator class ${\mathfrak{S}}_1.$ Thus, using the given above representation for the resolvent (5), we obtain

$$\Delta (\lambda ){\left({(I-\lambda B)}^{-1}f,f\right)}_{\mathfrak{H}}={\Delta }^{11}(\lambda ),$$

(18)

the latter entire function depends on the choice of an element $f\in \mathfrak{H}$ and we reflect this fact in the notation $\tilde{f}(\lambda ):={\Delta }^{11}(\lambda ).$ Let us notice the fact that $\tilde{f}(\lambda )$ represents the Fredholm determinant of the operator $Q_{1}B{Q}_1,$ where $Q_1$ is the projector into orthogonal complement of the element $f\in \mathfrak{H}.$ Having applied Lemma 2 (Lemma 1 [2]), we obtain

$$s_n(Q_{1}B{Q}_1)\le s_n(B),n\in \mathbb{N}.$$

Applying Lemma 3, we obtain the representation

$$\tilde{f}(\lambda )=\prod _{n=1}^{\infty }\left\{1-\lambda {\mu }_n(Q_{1}B{Q}_1)\right\},\lambda \in \mathbb{C}.$$

Applying Lemma 8, we have

$$\begin{gathered} |\tilde{f}(\lambda )|=\prod _{n=1}^{\infty }\left|1-\lambda {\mu }_n(Q_{1}B{Q}_1)\right|\le \prod _{n=1}^{\infty }|1+\lambda {\mu }_n(B)|\le \prod _{n=1}^{\infty }\left\{1+|\lambda {\mu }_n(B)|\right\}, \\ f\in \mathfrak{H},|arg\lambda |<\pi /4. \end{gathered}$$

(19)

Let us prove the following relation

$${|\Delta (\lambda )|·\parallel (I-\lambda B)}^{-1}\parallel \le 2\prod _{n=1}^{\infty }\left\{1+|\lambda {\mu }_n(B)|\right\},|arg\lambda |<\pi /4.$$

(20)

For this purpose, define an operator $D_B(\lambda ):=\Delta (\lambda ){(I-\lambda B)}^{-1}$ in accordance with (18), we have a correspondence between the notations ${(D_B(\lambda )f,f)}_{\mathfrak{H}}=\tilde{f}(\lambda ).$ Consider the decomposition on the Hermitian components

$$D_B(\lambda )=\mathfrak{Re}{D}_B(\lambda )+i\mathfrak{Im}{D}_B(\lambda ).$$

Note that the Hermitian components are selfadjoint operators. Thus, using the well-known formula for the norm of a selfadjoint operator, for an arbitrary fixed $\lambda \in \mathbb{C},$ we get

$$\parallel D_B(\lambda )\parallel =\underset{\parallel f\parallel \le 1}{sup}\parallel \mathfrak{Re}{D}_B(\lambda )f+i\mathfrak{Im}{D}_B{(\lambda )f\parallel }_{\mathfrak{H}}\le \underset{\parallel f\parallel \le 1}{sup}\parallel \mathfrak{Re}{D}_B{(\lambda )f\parallel }_{\mathfrak{H}}+\underset{\parallel f\parallel \le 1}{sup}{\parallel \mathfrak{Im}{D}_B(\lambda )f\parallel }_{\mathfrak{H}}=$$

$$=\underset{\parallel f\parallel =1}{sup}|{(\mathfrak{Re}{D}_B(\lambda )f,f)}_{\mathfrak{H}}|+\underset{\parallel f\parallel =1}{sup}|{(\mathfrak{Im}{D}_B(\lambda )f,f)}_{\mathfrak{H}}|=$$

$$=\underset{\parallel f\parallel =1}{sup}|\mathrm{Re}{(D_B(\lambda )f,f)}_{\mathfrak{H}}|+\underset{\parallel f\parallel =1}{sup}|\mathrm{Im}{(D_B(\lambda )f,f)}_{\mathfrak{H}}|\le 2\underset{\parallel f\parallel =1}{sup}|{(D_B(\lambda )f,f)}_{\mathfrak{H}}|=2\underset{\parallel f\parallel =1}{sup}|\tilde{f}(\lambda )|.$$

Taking into account inequality (19), we obtain (20). In accordance with the made assumptions, we have $n(r,B)=o(r^s),0<c<s<1,$ hence

$$\sum _{n=1}^{\infty }|{\mu }_n(B)|<\infty ,$$

therefore, applying Lemma 1 to the canonical product, we obtain

$${|\Delta (\lambda )|·\parallel (I-\lambda B)}^{-1}\parallel \le 2\prod _{n=1}^{\infty }\left\{1+|\lambda {\mu }_n(B)|\right\}\le 2e^{h(|\lambda |)},|arg\lambda |<\pi /4.$$

Now, to obtain the lemma statement it suffices to estimate the absolute value of the Fredholm determinant of the operator B from below. For this purpose, let us notice that in accordance with Lemma 3 it is an entire function represented by the formula

$$\Delta (\lambda )=\prod _{n=1}^{\infty }\left\{1-\lambda {\mu }_n(B)\right\},\lambda \in \mathbb{C}.$$

In accordance with the Joseph Cartan concept, we can obtain an estimate from below for the entire function that holds on the complex plane except may be an exceptional set of circulus. The latter cannot be found but and we are compelled to make an exclusion evaluating the measures. However, in the paper [42], we produce the method allowing to find exceptional set of circulus. In the simplified case, we can use Theorem 4 [38] (p. 79) giving the lower bound of the absolute value for an analytic function in the disk, we have

$$ln|\Delta (\lambda )|\ge -ln\left\{\underset{\psi \in [0,2\pi ]}{max}|\Delta (2eR{e}^{i\psi })|\right\}ln{\displaystyle \frac{15e^3}{\eta }},|\lambda |\le R,$$

except for the exceptional set of circles with the sum of radii less that $\eta R,$ where $\eta$ is an arbitrary small positive number. Thus, to find the desired circle $\lambda =e^{i\psi }\tilde{R}$ belonging to the ring, i.e., $R(1-\delta )<\tilde{R}<R,$ we have to choose $\eta$ satisfying the inequality

$$2\eta R<R-R(1-\delta )=\delta R;\eta <\delta /2.$$

Hence, having chosen $\eta =\delta /3,$ we get

$$ln|\Delta (\lambda )|\ge -ln\left\{\underset{\psi \in [0,2\pi ]}{max}|\Delta (2eR{e}^{i\psi })|\right\}ln{\displaystyle \frac{45e^3}{\delta }},|\lambda |=\tilde{R},$$

Analogously to the above, applying Lemma 1 to the canonical product, we get

$$|\Delta (\lambda )|=\left|\prod _{n=1}^{\infty }\left\{1-\lambda {\mu }_n(B)\right\}\right|\le e^{h(|\lambda |)}.$$

Therefore

$$ln\left\{\underset{\psi \in [0,2\pi ]}{max}|\Delta (2eR{e}^{i\psi })|\right\}\le h(2eR).$$

Substituting, we get

$$ln|\Delta (\lambda )|\ge -h\left(2eR\right)ln{\displaystyle \frac{45e^3}{\delta }};|\Delta (\lambda )|\ge e^{-h\left(2eR\right)ln{\displaystyle \frac{45e^3}{\delta }}},|\lambda |=\tilde{R}.$$

Note that

$${\displaystyle \frac{dh}{dr}}=\underset{r}{\overset{\infty }{\int }}{\displaystyle \frac{n(t,B)dt}{t^2}}>0.$$

(21)

Hence, using the monotonous property of the function $h(r),$ we conclude that $h(\tilde{R})<h\left(2eR\right).$ Combining the upper and the lower estimates, we obtain

$${\parallel (I-\lambda B)}^{-1}\parallel ={\displaystyle \frac{\parallel D_B(\lambda )\parallel }{|\Delta (\lambda )|}}\le 2e^{\left(1+ln{\displaystyle \frac{45e^3}{\delta }}\right)h\left(2eR\right)},|\lambda |=\tilde{R},|arg\lambda |<\pi /4.$$

The proof is complete. □

3.4. Infinitesimalness of the Summation Order

The following theorem is formulated in terms of Section 3.2.

Theorem 1.

Assume that $B\in {\mathfrak{S}}_{\sigma },0<\sigma \le 1,\Theta (B)\subset {\mathfrak{L}}_0(\theta ),\theta <\pi /4,$ has the sequence of the algebraic multiplicities of the ϕ-growth, then

$$B\in \mathcal{A}(s,\mathrm{R}(B)),\sigma \varphi <s<\pi /2\theta .$$

Proof. 

Firstly, let us note that since $iB$ is a dissipative operator, then in accordance with Theorem 2.3 paragraph 2 Chapter V [36], the system of the root vectors of the operator (including the root vectors corresponding to the zero eigenvelue) is complete in $\mathfrak{H}.$ Using notations of Section 3.2, having chosen an arbitrary small $\beta >0$ let us rearrange the sequence of the characteristic numbers ${\{{\lambda }_n\}}_1^{\infty }$ of the operator B in the groups ${\{{\lambda }_{k_j}\}}_1^{\infty }$ in accordance with (12). Applying Lemma 6, we can put the operators $B_k$ into correspondence with the sequences ${\{{\lambda }_{k_j}\}}_1^{\infty }.$ In accordance with Lemma 7, we have

$$\underset{r\to \infty }{lim}{\displaystyle \frac{n(r,B_k)}{r^s}}=0,k\in {\mathbb{N}}_0,s>\sigma \left({\displaystyle \frac{\beta }{\gamma }}+\varphi \right),\gamma :=\beta +1.$$

Consider orhtogonal projectors $P_k$ corresponding to the invariant subspaces ${\mathfrak{M}}_k$ defined in Lemma 6. It is clear that $B_k=P_{k}B{P}_k.$ Applying Lemma 2, we get

$$s_n(B_k)\le s_n(B),n\in \mathbb{N}.$$

Therefore $B_k\in {\mathfrak{S}}_1,k\in {\mathbb{N}}_0.$ Define a contour in the complex plane

$$\begin{gathered} \vartheta :=\left\{z\in \mathbb{C}:|z|=r_0,|\arg z|\le {\theta }_0\right\}\cup \left\{z\in \mathbb{C}:|z|>r_0,|\arg z|={\theta }_0\right\}, \\ {r}_0:=|{\lambda }_1|-\varsigma ,{\theta }_0:=\theta +\varsigma , \end{gathered}$$

(22)

where $\varsigma$ is an arbitrary small positive fixed number. Now consider a sequence of the radii $R_{\mu }=a\mu +b,\mu \in {\mathbb{N}}_0,a>b=r_0$ and define ${\delta }_{\mu }$ from the condition $R_{\mu }(1-{\delta }_{\mu })=a\mu ,$ we have

$${\delta }_{\mu }={\displaystyle \frac{1}{1+\mu a/b}},R_{\mu +1}(1-{\delta }_{\mu +1})>R_{\mu },$$

additionally without loss of generality, we assume that ${\{z_j\}}_1^{\infty }\cap {\{R_{\mu }\}}_0^{\infty }=\varnothing .$ Applying Lemma 9, we obtain the fact that in each case there exists a sequence of contours ${\{{\tilde{R}}_{k\mu }\}}_0^{\infty },R_{\mu }(1-{\delta }_{\mu })<{\tilde{R}}_{k\mu }<R_{\mu }$ such that the estimates hold

$$\parallel {(I-\lambda B_k)}^{-1}\parallel \le 2e^{h_k(2e{R}_{\mu })ln{\displaystyle \frac{45e^4}{{\delta }_{\mu }}}},|\lambda |={\tilde{R}}_{k\mu },|arg\lambda |<\pi /4,k\in {\mathbb{N}}_0,$$

where

$$h_k(r):=\underset{0}{\overset{r}{\int }}{\displaystyle \frac{n(t,B_k)dt}{t}}+r\underset{r}{\overset{\infty }{\int }}{\displaystyle \frac{n(t,B_k)dt}{t^2}}.$$

Additionally, we can assume that ${\{z_j\}}_1^{\infty }\cap {\tilde{R}}_{k\mu }=\varnothing ,k,\mu \in {\mathbb{N}}_0$ by virtue of arbitrariness in the choice of ${\tilde{R}}_{k\mu }$ dictated by Theorem 4 [38] (p. 79). Estimating, we get

$$\parallel {(I-\lambda B_k)}^{-1}\parallel \le 2e^{h_k(2e{R}_{\mu })ln[45e^4(1+\mu a/b)]}\le e^{C_0{h}_k(2e{R}_{\mu })ln\mu },|\lambda |={\tilde{R}}_{k\mu },|arg\lambda |<\pi /4,k\in {\mathbb{N}}_0.$$

(23)

Let us prove that

$$\underset{r\to \infty }{lim}{\displaystyle \frac{h_k(r)}{r^s}}=0,r\to \infty ,$$

(24)

uniformly with respect to $k\in {\mathbb{N}}_0.$ Without loss of generality, we assume that $s<1,$ applying Lemma 7, estimating the counting function under the integrals, we get

$$\forall \epsilon >0,\exists M(\epsilon ):h_k(r)<\epsilon \left\{\underset{0}{\overset{r}{\int }}{t}^{s-1}dt+r\underset{r}{\overset{\infty }{\int }}{t}^{s-2}dt\right\}={\displaystyle \frac{\epsilon r^s}{s(1-s)}},r>M(\epsilon ),k\in {\mathbb{N}}_0,$$

what means the desired result (24). Define a subsequence of the natural numbers ${\{M_{\mu }\}}_0^{\infty }$ as follows a number $M_{\mu }$ indicates a quantity of principal characteristic numbers of the operator B belonging to the open disk with the radius $R_{\mu },$ i.e.,

$$M_{\mu }:=\mathrm{card}\{j\in \mathbb{N}:|z_j|<R_{\mu }\}.$$

Then the group $z_{M_{\mu }+1},z_{M_{\mu }+2},\dots ,z_{M_{\mu +1}}$ of the principal characteristic numbers is inclosed in the closed contour formed by the intersection of the contour $\vartheta$ with the circulus having radii $R_{\mu },R_{\mu +1}.$ Following to the Lidskii V.B. [2] results consider the following relation

$${\displaystyle \frac{1}{2\pi i}}\underset{\vartheta (R_{m+1})}{\oint }{e}^{-{\lambda }^{s}t}B{\left(I-\lambda B\right)}^{-1}fd\lambda =-\sum _{\mu =0}^m\sum _{q=M_{\mu }+1}^{M_{\mu +1}}{\mathcal{P}}_q(s,t)f,f\in \mathfrak{H},m\in \mathbb{N},$$

where $\sigma \varphi <s<\pi /2\theta ,$

$$\vartheta (R_{m+1}):=\left\{z\in \mathbb{C}:|z|=r_0,|z|=R_{m+1},|\arg z|\le {\theta }_0\right\}\cup \left\{z\in \mathbb{C}:r_0<|z|<R_{m+1},|\arg z|={\theta }_0\right\}.$$

Observe that the inner sum contains $M_{\mu +1}-M_{\mu }$ terms, we have

$$M_{m+1}=\sum _{\mu =0}^m{M}_{\mu +1}-M_{\mu }.$$

The Lidskii V.B. idea is to pass to the limit in the last integral when m tends to infinity and in this way to prove the series convergence. However, there are some obstacles in evaluating the norm of the integral, we should take a value of the summation order more then the index of the Schatten-von Neumann class. At the same time the heuristic reasonings lead us to the hypotheses that the decay of the exponential function has a surplus, therefore the latter can be replaced in the construction or at least the infinitesimal value of the order can be considered. Consider a sum

$$\sum _{k=0}^{\psi (\mu )}\sum _{j=M_{k\mu }+1}^{M_{k\mu +1}}{\mathcal{P}}_{k_j}(s,t)f,$$

where the operators in the inner sum correspond to the principal characteristic numbers $z_{k_j},j=M_{k\mu }+1,M_{k\mu }+2,\dots ,M_{k\mu +1}$ (see Section 3.2), the number $M_{k\mu }$ indicates a quantity of principal characteristic numbers of the operator $B_k$ belonging to the open disk with the radius ${\tilde{R}}_{k\mu },$ i.e.,

$$M_{k\mu }:=\mathrm{card}\{j\in \mathrm{N},|z_{k_j}|<{\tilde{R}}_{k\mu }\}.$$

The symbol $\psi (\mu ):=\mathrm{card}\{k\in {\mathbb{N}}_0:|z_{k_j}|<{\tilde{R}}_{k\mu +1}\}$ denotes a function indicating a quantity of operators $B_k$ having characteristic numbers inside the circles with the radii ${\tilde{R}}_{k\mu +1}.$ Observe the representation

$$\sum _{k=0}^{\psi (m)}{M}_{km+1}=\sum _{k=0}^{\psi (m)}\sum _{\mu =0}^m\{M_{k\mu +1}-M_{k\mu }\}=\sum _{\mu =0}^m\sum _{k=0}^{\psi (\mu )}\{M_{k\mu +1}-M_{k\mu }\},$$

here we have taken into account the fact $M_{k\mu +1}=0,k>\psi (\mu ).$ Thus, if we analyze a mutual arrangement of the radii $R_{\mu },{\tilde{R}}_{k\mu },$ we come to the fact that there exists a natural number m such that

$$M_{m+1}\ne \sum _{k=0}^{\psi (m)}{M}_{km+1}.$$

The latter relation can be rewritten in terms of operators

$$\sum _{\mu =0}^m\sum _{q=M_{\mu }+1}^{M_{\mu +1}}{\mathcal{P}}_q(s,t)f\ne \sum _{\mu =0}^m\sum _{k=0}^{\psi (\mu )}\sum _{j=M_{k\mu }+1}^{M_{k\mu +1}}{\mathcal{P}}_{k_j}(s,t)f,f\in \mathfrak{H}.$$

However, we can produce a subsequence ${\{{\xi }_l\}}_1^{\infty }\subset \mathbb{N}$ so that that the left-hand side and the right-hand side became equal if $m={\xi }_l,l\in \mathbb{N}.$ Note that in accordance with (14), we have

$$B\in {\mathfrak{S}}_1\Rightarrow |{\mu }_n(B)|=o\left(n^{-1}\right),$$

therefore $n(r,B)\le \epsilon r,$ for sufficiently large values $r,$ where $\epsilon >0$ is an arbitrary small positive number. Consider a counting function corresponding to the sequence ${\{R_{\mu }\}}_0^{\infty },$ it is clear that $n(r,R_{\mu })=a^{-1}r+o(r).$ Consider the difference

$$n(r,R_{\mu })-n(r,B)\ge \left\{a^{-1}-\epsilon \right\}r+o(r),$$

(25)

it is clear that $n(r,R_{\mu })-n(r,B)\to \infty ,r\to \infty .$ Thus, we can extract a subsequence of the natural numbers ${\{{\xi }_l\}}_1^{\infty }\subset \mathbb{N}$ so that the sequence ${\left\{n(R_{{\xi }_l+1},R_{\mu })-n(R_{{\xi }_l+1},B)\right\}}_1^{\infty }$ is monotonically increasing. Therefore each ring $\left\{z\in \mathbb{C}:R_{{\xi }_l}<|z|<R_{{\xi }_l+1}\right\}$ does not contain characteristic numbers of the operator $B.$ Now if we consider possible arrangements of the radii ${\tilde{R}}_{k\mu }$ then the following fact becomes clear

$$M_{{\xi }_l+1}=\sum _{k=0}^{\psi ({\xi }_l)}{M}_{k{\xi }_l+1},l=1,2,\dots .$$

Therefore

$${\displaystyle \frac{1}{2\pi i}}\underset{\vartheta (R_{{\xi }_l+1})}{\oint }{e}^{-{\lambda }^{s}t}B{\left(I-\lambda B\right)}^{-1}fd\lambda =-\sum _{\mu =0}^{{\xi }_l}\sum _{q=M_{\mu }+1}^{M_{\mu +1}}{\mathcal{P}}_q(s,t)f=$$

$$=-\sum _{\mu =0}^{{\xi }_l}\sum _{k=0}^{\psi (\mu )}\sum _{j=M_{k\mu }+1}^{M_{k\mu +1}}{\mathcal{P}}_{k_j}(s,t)f,f\in \mathfrak{H},l\in \mathbb{N}.$$

Apparently, if we prove the fact

$$\sum _{\mu =0}^{\infty }\sum _{k=0}^{\psi (\mu )}{∥\sum _{j=M_{k\mu }+1}^{M_{k\mu +1}}{\mathcal{P}}_{k_j}(s,t)f∥}_{\mathfrak{H}}<\infty ,$$

then we obtain

$${\displaystyle \frac{1}{2\pi i}}\underset{\vartheta }{\int }{e}^{-{\lambda }^{s}t}R(\lambda )fd\lambda =\sum _{\mu =0}^{\infty }\sum _{q=M_{\mu }+1}^{M_{\mu +1}}{\mathcal{P}}_q(s,t)f,$$

(26)

where the integration direction is chosen so that the inside of the domain containing the real axis appears at the right-hand side while the point is going along the contour, moreover

$$\sum _{l=1}^{\infty }{∥\sum _{\mu ={\xi }_l}^{{\xi }_{l+1}}\sum _{q=M_{\mu }+1}^{M_{\mu +1}}{\mathcal{P}}_q(s,t)f∥}_{\mathfrak{H}}<\infty .$$

For this purpose let us estimate $\psi (\mu ),$ observe the following relation

$$\psi (\mu )\le \mathrm{card}\{\nu \in {\mathbb{N}}_0:|z_{N_{\nu }}|\le R_{\mu +1}\}+1<\mathrm{card}\{j\in \mathbb{N}:|{\lambda }_j|\le R_{\mu +1}\}+1=$$

$$=n(R_{\mu +1},B)+1<n(R_{\mu +1},R_{\mu })+1=\mu +2,$$

it holds for a sufficiently large value $\mu ,$ in accordance with (25). Therefore, if we prove that the following series is convergent, i.e.,

$$S\le \sum _{\mu =0}^{\infty }\sum _{k=0}^{\mu +2}{∥\sum _{j=M_{k\mu }+1}^{M_{k\mu +1}}{\mathcal{P}}_{k_j}(s,t)f∥}_{\mathfrak{H}}<\infty ,$$

then we obtain the desired result. Since the inner sum contains the projectors corresponding to the operator $B_k$ then we can apply Lidskii V.B. method [2] modified by virtue of Lemma 9. Let us estimate the inner sum, we have

$${∥\sum _{j=M_{k\mu }+1}^{M_{k\mu +1}}{\mathcal{P}}_{k_j}(s,t)f∥}_{\mathfrak{H}}={∥\underset{{\vartheta }_{k\mu }}{\int }{e}^{-{\lambda }^{s}t}{B}_k{\left(I-\lambda B_k\right)}^{-1}fd\lambda ∥}_{\mathfrak{H}}\le J_{k\mu }+J_{k\mu +1}+J_{k\mu }^{+}+J_{k\mu }^{-},$$

$$J_{k\mu }:={∥\underset{{\tilde{\vartheta }}_{k\mu }}{\int }{e}^{-{\lambda }^{s}t}{B}_k{\left(I-\lambda B_k\right)}^{-1}fd\lambda ∥}_{\mathfrak{H}},J_{k\mu }^{+}:={∥\underset{{\vartheta }_{k{\mu }_{+}}}{\int }{e}^{-{\lambda }^{s}t}{B}_k{\left(I-\lambda B_k\right)}^{-1}fd\lambda ∥}_{\mathfrak{H}},$$

$$J_{k\mu }^{-}:={∥\underset{{\vartheta }_{k{\mu }_{-}}}{\int }{e}^{-{\lambda }^{s}t}{B}_k{\left(I-\lambda B_k\right)}^{-1}fd\lambda ∥}_{\mathfrak{H}},$$

$${\tilde{\vartheta }}_{k\mu }:=\{z\in \mathbb{C}:|z|={\tilde{R}}_{k\mu },|argz|\le {\theta }_0\},{\vartheta }_{k{\mu }_{\pm }}:=\{z\in \mathbb{C}:{\tilde{R}}_{k\mu }<|z|<{\tilde{R}}_{k\mu +1},argz=\pm {\theta }_0\}.$$

$${\vartheta }_{k\mu }={\tilde{\vartheta }}_{k\mu }\cup {\vartheta }_{k{\mu }_{+}}\cup {\tilde{\vartheta }}_{k\mu +1}\cup {\vartheta }_{k{\mu }_{-}}.$$

Applying (23), we get

$$J_{k\mu }\le \underset{{\tilde{\vartheta }}_{k\mu }}{\int }{e}^{-t\mathrm{Re}{\lambda }^s}{∥B_k{\left(I-\lambda B_k\right)}^{-1}f∥}_{\mathfrak{H}}|d\lambda |\le C\parallel f\parallel e^{C_0{h}_k(2e{R}_{\mu })ln\mu }{\tilde{R}}_{k\mu }\underset{-{\theta }_0}{\overset{{\theta }_0}{\int }}{e}^{-t\mathrm{Re}{\lambda }^s}d\arg \lambda ,|\lambda |={\tilde{R}}_{k\mu }.$$

Since in accordance with the made assumptions, we have ${\theta }_0<\pi /2s$ then

$$\mathrm{Re}{\lambda }^s\ge {|\lambda |}^{s}cos{\theta }_0{s>|\lambda |}^{s}cos\left[(\pi /2s-\epsilon )s\right]={|\lambda |}^{s}sin\epsilon s,\lambda \in {\vartheta }_{k\mu },$$

where $\epsilon$ is a sufficiently small positive value. Using this estimate, we get

$$ln{J}_{k\mu }\le lnC+C_0{h}_k(2e{R}_{\mu })ln\mu -{\tilde{R}}_{k\mu }^{s}tsin\epsilon s\le lnC+C_0{h}_k(2e{R}_{\mu })ln\mu -R_{\mu -1}^{s}tsin\epsilon s=$$

$$=lnC+R_{\mu -1}^s\left\{C_0{R}_{\mu -1}^{-s}{h}_k(2e{R}_{\mu })ln\mu -tsin\epsilon s\right\},k\in {\mathbb{N}}_0.$$

Using (24), we get $R_{\mu -1}^{-s}{h}_k(2e{R}_{\mu })ln\mu \to 0,\mu \to \infty ,$ uniformly with respect to $k\in {\mathbb{N}}_0.$ Therefore

$$ln{J}_{k\mu }\le C{e}^{-C_1{R}_{\mu }^s}.$$

To estimate other terms, we are rather satisfied with the estimate represented in Lemma 4 (Lidskii V.B.) [2], what gives us the following relation

$$\parallel {(I-\lambda B_k)}^{-1}\parallel \le {\displaystyle \frac{1}{sin(|\psi |-\theta )}},\lambda \in \{z\in \mathbb{C}:argz=\psi \},\theta <|\psi |<\pi /2.$$

Absolutely analogously to the reasonings represented in Lemma 7 [2], we get

$$J_{k\mu }^{\pm }\le {C\parallel f\parallel }_{\mathfrak{H}}\underset{{\tilde{R}}_{k\mu }}{\overset{{\tilde{R}}_{k\mu +1}}{\int }}{e}^{-t\mathrm{Re}{\lambda }^s}|d\lambda |\le C\underset{R_{\mu -1}}{\overset{\infty }{\int }}{e}^{-t{r}^{s}sins\epsilon }dr=s^{-1}\underset{R_{\mu -1}^s}{\overset{\infty }{\int }}{e}^{-t\phi sins\epsilon }{\phi }^{{\displaystyle \frac{1}{s}}-1}d\phi \le {\displaystyle \frac{e^{-t{R}_{\mu -1}^{s}sins\epsilon }}{tsins\epsilon }}.$$

Thus, we have come to the relation

$${∥\sum _{j=M_{k\mu }+1}^{M_{k\mu +1}}{\mathcal{P}}_{k_j}(s,t)f∥}_{\mathfrak{H}}\le C{e}^{-C_1{R}_{\mu }^s}+C{e}^{-C_1{R}_{\mu +1}^s}+2{\displaystyle \frac{e^{-t{R}_{\mu -1}^{s}sins\epsilon }}{tsins\epsilon }}.$$

Combining the obtained estimates, taking into account the fact $R_{\mu }^s\ge C{\mu }^s,$ we get

$$S\le \sum _{\mu =0}^{\infty }\sum _{k=0}^{\mu +2}{∥\sum _{j=M_{k\mu }+1}^{M_{k\mu +1}}{\mathcal{P}}_{k_j}(s,t)f∥}_{\mathfrak{H}}\le C\sum _{\mu =0}^{\infty }\sum _{k=0}^{\mu +2}{e}^{-C_2{\mu }^s}=\sum _{\mu =0}^{\infty }(\mu +3)e^{-C_2{\mu }^s}<\infty .$$

The convergence of the last series can be verified easily due to the integral test of convergence. Therefore, relation (26) holds. Applying Lemma 4, we complete the proof. □

3.5. Supplementary Remarks and Mathematical Applications

3.5.1. Evolution Equations

The obtained results admit the following application. By virtue of arbitrariness in choosing the summation order, we can find a solution analytically for the Cauchy problem for the fractional evolution equation containing in the second term an operator belonging to a sufficiently wide class, see (27). Some examples are represented in the paper [32], wherein fractional integro-differential operators generated by a $C_0$ semigroup of contractions such as the Riesz potential, the Riemann-Liouville fractional differential operator, the Kipriyanov operator, the difference operator are considered. General approach, realized in the paper [29], allows us to consider a special transform of m-accretive operator. This approach seems to be extremely relevant due to the fact that the class of m-accretive operators contains the infinitesimal generator of a $C_0$ semigroup of contractions. We should remark that a fractional differential operator of the real order can be expressed in terms of the infinitesimal generator of the corresponding semigroup, see [29]. Below, we represent an abstract scheme of possible applications.

Consider element-functions of the Hilbert space $u:{\mathbb{R}}_{+}\to \mathfrak{H},u:=u(t),t\ge 0$ assuming that if u belongs to $\mathfrak{H}$ then the fact holds for all values of the variable $t.$ Notice that under such assumptions 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 a limit

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

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

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

where $(a=t_0<t_1<\dots <t_n=b)$ is an arbitrary splitting of the segment $J,\zeta :=\underset{i}{max}(t_{i+1}-t_i),{\xi }_i$ is an arbitrary point belonging to $[t_i,t_{i+1}].$ The sufficient condition of the last integral existence is a continuous property (see [43] (p. 248)), i.e., $u(t)\stackrel{\mathfrak{H}}{⟶}u(t_0),t\to t_0,\forall t_0\in J.$ The improper integral is understood as a limit

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

Consider a fractional integral in the Riemann-Liouvile sense (see [44])

$$I_{-}^{\alpha }f(t)={\displaystyle \frac{1}{\Gamma (\alpha )}}\underset{0}{\overset{\infty }{\int }}f(t+x)x^{\alpha -1}dx,\alpha \ge 0.$$

Combining the generalized integro-differential operations, we can consider a fractional differential operator in the Riemann-Liouvile sense, in the formal form, we have

$${\mathfrak{D}}_{-}^{\alpha }f(t):={\displaystyle \frac{{(-1)}^n}{\Gamma (n-\alpha )}}{\displaystyle \frac{d^n}{d{t}^n}}\underset{0}{\overset{\infty }{\int }}f(t+x)x^{n-\alpha -1}dx,\alpha \ge 0,n=[\alpha ]+1.$$

Thus, we can write

$${\mathfrak{D}}_{-}^{\alpha }f(t)={(-1)}^n{\displaystyle \frac{d^n}{d{t}^n}}\left\{I_{-}^{n-\alpha }f(t)\right\}.$$

Here, we should remark that

$${\mathfrak{D}}_{-}^{n}f(t)={(-1)}^n{\displaystyle \frac{d^{n}f}{d{t}^n}},I_{-}^{n}f(t)=\underset{t}{\overset{\infty }{\int }}d{x}_1\underset{x_1}{\overset{\infty }{\int }}d{x}_2\dots \underset{x_{n-1}}{\overset{\infty }{\int }}f(x_n)d{x}_n,n\in \mathbb{N}.$$

In accordance with the accepted unified form of notation, that stresses the inverse nature of the operators, we can write

$${\mathfrak{D}}_{-}^{-\alpha }f(t)=I_{-}^{\alpha }f(t),\alpha \in \mathbb{R}.$$

Throughout this paragraph, we consider an operator with discrete spectrum W satisfying the conditions $\Theta (W)\in {\mathfrak{L}}_0(\theta ),\theta <\pi /2.$ Applying the reasonings represented in Lemma 5, it is not hard to prove that the operator W is m-accretive. Therefore, using relations (3.41), (3.53) [34], we can define fractional powers of the operator W as follows

$$W^{-\beta }f={\displaystyle \frac{1}{2\pi i}}\underset{\vartheta }{\int }{\lambda }^{-\beta }{\left(W-\lambda I\right)}^{-1}fd\lambda ,W^{\beta }f={\displaystyle \frac{1}{2\pi i}}\underset{\vartheta }{\int }{\lambda }^{\beta -1}{\left(W-\lambda I\right)}^{-1}Wfd\lambda ,$$

$$f\in \mathrm{D}(W),\beta \in (0,1),$$

where the contour $\vartheta$ is defined in (22) Let us study a Cauchy problem

$${\mathfrak{D}}_{-}^{\alpha }u(t)=Wu(t),\alpha >0,$$

(27)

with the initial conditions

$$\underset{t\to 0}{lim}{\mathfrak{D}}_{-}^{k+\alpha -n}u(t)=W^{{\alpha }_k}h,h\in \mathrm{D}(W),$$

$${\alpha }_k=(\{\alpha \}+k-1)/\alpha ,k=\left\{\begin{array}{c}\hfill 0,1,\dots ,n-1,\{\alpha \}\ne 0\\ \hfill 1,2,\dots ,n-1,\{\alpha \}=0\end{array}\right..$$

Thus, in the case corresponding to $\alpha =1$ the Cauchy problem can be rewritten in the classical form

$${\displaystyle \frac{du}{dt}}=-Wu,\underset{t\to 0}{lim}u(t)=h\in \mathrm{D}(W).$$

This case was studied by Lidskii V.B. in the paper [2], under the assumption $B\in {\mathfrak{S}}_{\sigma },\sigma \le 1$ as the most relevant application of the method $(A,\lambda ,s).$ In the case $\alpha =2$ the Cauchy problem can be rewritten in the form

$${\displaystyle \frac{d^{2}u}{d{t}^2}}=Wu,\underset{t\to 0}{lim}u(t)=h,\underset{t\to 0}{lim}{\displaystyle \frac{du}{dt}}=-\sqrt{W}h,h\in \mathrm{D}(W).$$

However, the principal result obtained in this paper allows to consider higher orders of fractional derivatives independently on the Schatten-von Neumann index what is reflected in the following theorem.

Theorem 2.

Assume that the operator $B:=W^{-1}$ satisfies conditions of Theorem 1, $2\theta /\pi <\alpha <1/\sigma \varphi ,$ then there exists a solution of the Cauchy problem (27) in the form

$$u(t)=\sum _{\mu =0}^{\infty }\sum _{q=M_{\mu }+1}^{M_{\mu +1}}{\mathcal{P}}_q({\alpha }^{-1},t)h.$$

(28)

Proof. 

Consider an element-function

$$u(t):={\displaystyle \frac{1}{2\pi i}}\underset{\vartheta }{\int }{e}^{-{\lambda }^{1/\alpha }t}B{\left(I-\lambda B\right)}^{-1}hd\lambda ,t>0.$$

Note that in accordance with Theorem 1 relation (28) holds. Thus, we should prove the fact that $u(t)$ is a solution of the equation satisfying the initial conditions. In accordance with the preliminary information given above, we can write

$${\mathfrak{D}}_{-}^{k+\alpha -n}u(t)={(-1)}^k{\displaystyle \frac{d^k}{d{t}^k}}\left\{I_{-}^{n-\alpha }u(t)\right\},k=0,1,\dots ,n.$$

Changing the order of integration, what is based on the statements of the ordinary calculus, we get

$$\Gamma (n-\alpha )I_{-}^{n-\alpha }u(t)={\left(\underset{0}{\overset{\infty }{\int }}{x}^{[\alpha ]-\alpha }u(t+x)dx,g\right)}_{\mathfrak{H}}={\displaystyle \frac{1}{2\pi i}}\underset{\vartheta }{\int }{e}^{-{\lambda }^{1/\alpha }(t+x)}{\left(R(\lambda )h,g\right)}_{\mathfrak{H}}d\lambda \underset{0}{\overset{\infty }{\int }}{x}^{[\alpha ]-\alpha }dx=$$

$$={\displaystyle \frac{1}{2\pi i}}\underset{0}{\overset{\infty }{\int }}{x}^{[\alpha ]-\alpha }{e}^{-{\lambda }^{1/\alpha }x}dx\underset{\vartheta }{\int }{e}^{-{\lambda }^{1/\alpha }t}{\left(R(\lambda )h,g\right)}_{\mathfrak{H}}d\lambda ,g\in \mathfrak{H},$$

where $R(\lambda )=B{(I-\lambda B)}^{-1}.$ Notice that

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

therefore

$$I_{-}^{n-\alpha }u(t)={\displaystyle \frac{1}{2\pi i}}\underset{\vartheta }{\int }{e}^{-{\lambda }^{1/\alpha }t}{\lambda }^{1-n/\alpha }R(\lambda )hd\lambda .$$

Hence

$${\mathfrak{D}}_{-}^{k+\alpha -n}u(t)={\displaystyle \frac{{(-1)}^k}{2\pi i}}{\displaystyle \frac{d^k}{d{t}^k}}\underset{\vartheta }{\int }{e}^{-{\lambda }^{1/\alpha }t}{\lambda }^{1-n/\alpha }R(\lambda )hd\lambda ={\displaystyle \frac{1}{2\pi i}}\underset{\vartheta }{\int }{e}^{-{\lambda }^{1/\alpha }t}{\lambda }^{1-(n-k)/\alpha }R(\lambda )hd\lambda ,$$

$$k=0,1,\dots ,n.$$

The differentiation under the integral can be easily substantiated analogously to the proposition related to the ordinary calculus, the detailed reasonings are represented in the proof of Theorem 4 [42]. Taking into account the fact $\lambda R(\lambda )={(I-\lambda B)}^{-1}-I,$ we get

$${\mathfrak{D}}_{-}^{k+\alpha -n}u(t)={\displaystyle \frac{1}{2\pi i}}\underset{\vartheta }{\int }{e}^{-{\lambda }^{1/\alpha }t}{\lambda }^{1-(n-k)/\alpha }R(\lambda )hd\lambda =$$

$$={\displaystyle \frac{1}{2\pi i}}\underset{\vartheta }{\int }{e}^{-{\lambda }^{1/\alpha }t}{\lambda }^{-(n-k)/\alpha }{(I-\lambda B)}^{-1}hd\lambda +{\displaystyle \frac{h}{2\pi i}}\underset{\vartheta }{\int }{e}^{-{\lambda }^{1/\alpha }t}{\lambda }^{-(n-k)/\alpha }d\lambda .$$

Note that the second integral equals to zero due to the analytic property of the subintegral function in the domain

$$\left\{z\in \mathbb{C}:|z|=r_0,|z|=R,|argz|\le {\theta }_0\right\}\cup \left\{z\in \mathbb{C}:r_0<|z|<R,|argz|={\theta }_0\right\},$$

where $R>0$ is an arbitrary large number, and the fact

$$\underset{{\vartheta }_R}{\int }{e}^{-{\lambda }^{1/\alpha }t}{\lambda }^{-(n-k)/\alpha }d\lambda \to 0,R\to \infty ,{\vartheta }_R:=\left\{z\in \mathbb{C}:|z|=R,|argz|\le {\theta }_0\right\}.$$

Therefore

$${\mathfrak{D}}_{-}^{k+\alpha -n}u(t)={\displaystyle \frac{1}{2\pi i}}\underset{\vartheta }{\int }{e}^{-{\lambda }^{1/\alpha }t}{\lambda }^{-(n-k)/\alpha }{(I-\lambda B)}^{-1}hd\lambda ={\displaystyle \frac{1}{2\pi i}}\underset{\vartheta }{\int }{e}^{-{\lambda }^{1/\alpha }t}{\lambda }^{-(n-k)/\alpha }R(\lambda )Whd\lambda =$$

$$={\displaystyle \frac{1}{2\pi i}}\underset{\vartheta }{\int }{e}^{-{\lambda }^{1/\alpha }t}{\lambda }^{-(n-k)/\alpha }WR(\lambda )hd\lambda ,$$

(29)

from what follows the fact ${\mathfrak{D}}_{-}^{\alpha }u(t)=Wu(t),$ if we put $k=n.$ Assume that k corresponds to the initial conditions. Observe that the commutative property $B{(I-\lambda B)}^{-1}={(I-\lambda B)}^{-1}B,$ see Problem 5.4 [34] (p. 36), leads to the equality ${(W-\lambda I)}^{-1}=R(\lambda ).$ It follows that that the improper integral (29) is uniformly convergent with respect to the parameter t since we have

$$\parallel R(\lambda )\parallel \le {C|\lambda |}^{-1},\lambda \in \{z\in \mathbb{C}:argz=\psi \},\theta <|\psi |<\pi /2,$$

see the proof of Lemma 5. Therefore, passing to the limit under the integral, we get

$${\mathfrak{D}}_{-}^{\alpha -n}u(t)\to {\displaystyle \frac{1}{2\pi i}}\underset{\vartheta }{\int }{\lambda }^{1-n/\alpha }R(\lambda )hd\lambda =W^{1-n/\alpha }h,t\to 0,\{\alpha \}\ne 0,$$

$${\mathfrak{D}}_{-}^{k+\alpha -n}u(t)\to {\displaystyle \frac{1}{2\pi i}}\underset{\vartheta }{\int }{\lambda }^{-(n-k)/\alpha }R(\lambda )Whd\lambda =W^{1-(n-k)/\alpha }h,t\to 0,$$

$$k=1,2,\dots ,n-1,$$

where the last formula is true except for the case $\{\alpha \}=0,k=1$ which is covered by Lemma 5. The proof is complete. □

The main advantage of the last theorem is that there are not any restrictions upon the highest order of the fractional derivative or the index of the Schatten-von Neumann class in the case when the operator $W^{-1}$ has the sequence of the algebraic multiplicities of the lowest growth. It follows from the opportunity to consider an arbitrary small value of the summation order in accordance with Theorem 1.

3.5.2. Spectral Asymptotics for Fractional-Differential and Pseudo-Differential Operators

In this paragraph, we study an operator with discrete spectrum $W.$ Consider a formula obtained by Markus A.S., Matsaev V.I. (2.1) [15] connecting spectral asymptotic of the operator and its real component. Assume that condition

$${|(\mathfrak{Im}Wf,f)}_{\mathfrak{H}}|\le {\displaystyle \frac{b}{2}}\parallel \sqrt{\mathfrak{Re}W}f{\parallel }_{\mathfrak{H}}^{2q}{\parallel f\parallel }_{\mathfrak{H}}^{2-2q},f\in \mathrm{D}\left(\sqrt{\mathfrak{Re}W}\right),q\in [0,1),$$

(30)

holds then in accordance with Theorem 2.1 [15] for an arbitrary $\delta >0$ and $r>r_1$ the following relation holds

$$n(r,W)-n(r,\mathfrak{Re}W)\le K\left\{n\left(r+b[1+\delta ]r^q,\mathfrak{Re}W\right)-n\left(r-b[1+\delta ]r^q,\mathfrak{Re}W\right)\right\},$$

(31)

where K is a constant that depends on $\delta$ only, the constant $r_1$ depends of $\delta ,q,b.$ Here, we should remark that instead of condition (30) ($2^0$ in accordance with the terminology used in [15]), we can impose more general condition $3^0$ [15] that guarantees (31).

Consider a spectral asymptotics that is inherent to a wide class of fractional-differential and pseudo-differential operators, see [45]

$$n(r,\mathfrak{Re}W)={\gamma }_0{r}^{\xi }+o(r^{\mu }),r\to \infty ,0\le \mu <\xi .$$

Taking into account the fact obtained due the Taylor formula

$${(r+c{r}^q)}^{\xi }-{(r-c{r}^q)}^{\xi }=r^{\xi }\left\{{(1+c{r}^{q-1})}^{\xi }-{(1-c{r}^{q-1})}^{\xi }\right\}=2\xi c{r}^{\xi +q-1}+O(r^{\xi +2q-2}),$$

where $c=b[1+\delta ],$ substituting, we get

$$n(r,W)-n(r,\mathfrak{Re}W)\le K\left\{2\xi c{r}^{\xi +q-1}+o(r^{\mu })+O(r^{\xi +2q-2})\right\}.$$

Therefore $n(r,W)-n(r,\mathfrak{Re}W)=o(r^{\mu }),\mu >\xi +q-1.$ The given above reasonings lead us to the implication

$$n(r,\mathfrak{Re}W)={\gamma }_0{r}^{\xi }+o(r^{\mu }),\Rightarrow n(r,W)={\gamma }_0{r}^{\xi }+o(r^{\mu }),\mu >\xi +q-1.$$

(32)

However, we have an interest in a more explicit formula studied in [45,46]

$$n(r,\mathfrak{Re}W)=\sum _{j=0}^l{\gamma }_j{r}^{{\displaystyle \frac{v-j}{m}}}+o(r^{{\displaystyle \frac{v-l}{m}}}),0\le l\le v,$$

where $v\in \mathbb{N}$ is the dimension of the Euclidian space $m\in \mathbb{N}$ is the order of the operator (derivative), ${\gamma }_j$ are constants, ${\gamma }_0>0.$ Observe that by virtue of the assumption $l/m<1-q$ we are able to apply the scheme of reasonings used to obtain (32), analogously to the above, we get

$$n(r,\mathfrak{Re}W)=\sum _{j=0}^l{\gamma }_j{r}^{{\displaystyle \frac{v-j}{m}}}+o(r^{{\displaystyle \frac{v-l}{m}}}),\Rightarrow n(r,W)=\sum _{j=0}^l{\gamma }_j{r}^{{\displaystyle \frac{v-j}{m}}}+o(r^{{\displaystyle \frac{v-l}{m}}}).$$

(33)

This fact is noticed in Corollary 2.3 [15] for two-termed asymptotics. On the other hand, implementing the scheme of reasonings represented in [47] (p. 98) (6.1.15), we obtain

$$n(r,W)=\sum _{j=0}^l{\gamma }_j{r}^{{\displaystyle \frac{v-j}{m}}}+o(r^{{\displaystyle \frac{v-l}{m}}}),\Rightarrow |{\mu }_n(W)|=\sum _{j=0}^l{\tilde{\gamma }}_j{n}^{{\displaystyle \frac{m-j}{v}}}+o(n^{{\displaystyle \frac{m-l}{v}}}),$$

(34)

where ${\tilde{\gamma }}_j$ are constants ${\tilde{\gamma }}_0>0.$ Now, if we assume that $v=l,$ then $|{\mu }_{n+1}(W)|-|{\mu }_n(W)|>0$ for sufficiently large numbers $n\in \mathbb{N}.$ It follows that the operator $W^{-1}$ has the sequence of the algebraic multiplicities of the lowest growth. Summarizing the given above information, we conclude that the case $l=v,v/m<1-q$ falls within the scope of the developed method. Further, we illustrate the idea through the low values of l corresponding to well-known operators.

We have a particular interest in the case when condition (30) is satisfied and the following relation holds

$$n(r,\mathfrak{Re}W)={\gamma }_0{r}^{\xi }+O(lnr),0<\xi <1-q,$$

(35)

then in accordance with Corollary 2.2 [15], we get

$$n(r,W)={\gamma }_0{r}^{\xi }+O(lnr).$$

By direct calculations, we obtain

$$n^{a/\xi }(r)=r^a{\left\{{\gamma }_0+r^{-\xi }O(lnr)\right\}}^{a/\xi }={\gamma }_0^{a/\xi }{r}^a+r^{a-\xi }O(lnr),a>0,$$

where $n(r):=n(r,W),$ therefore

$${\displaystyle \frac{r^{1-\xi }O(lnr)}{n^{a/\xi }(r)}}\to 0,r\to \infty ,a>1-\xi .$$

Thus, we obtain

$$r={\gamma }_0^{-1/\xi }{n}^{1/\xi }(r)+r^{1-\xi }O(lnr);r={\gamma }_0^{-1/\xi }{n}^{1/\xi }+o(n^{a/\xi }),n\in \mathbb{N}.$$

(36)

Now assume that $\xi =v/m,1<v<m,$ then for the sake of certainty, we can choose a satisfying the condition

$$a/\xi =am/v=(m-1)/v>m/v-1=(1-\xi )/\xi ,$$

and rewrite relation (36) in the form

$$|{\mu }_n(W)|={\gamma }_0^{-1/\xi }{n}^{m/v}+o(n^{(m-1)/v}).$$

It follows that for sufficiently large values $j\in \mathbb{N},$ we have

$$|{\mu }_{{(j+1)}^v}(W)|-|{\mu }_{j^v}(W)|>0,$$

hence the principal indexes admit the following estimate $p_j\le C{j}^v.$ Consider a function $\Lambda (r):=n(r,W)-n(r-\delta ,W),$ where $\delta >0$ is an arbitrary small positive fixed number. It is clear that

$$\Lambda (r)\le C{r}^{v/m-1}+O(lnr).$$

Substituting (36), taking into account the fact $v<m,$ we obtain

$${\Delta }_j\le \Lambda ({\mu }_{p_j})\le C_1{p}_j^{1-m/v}+C_{2}ln{C}_3{p}_j\le Clnj,j\in \mathbb{N},$$

from what follows that the operator has the sequence of the algebraic multiplicities of the lowest growth. It is remarkable that conditions (33), (35) allow to establish the fact that the summation order is an arbitrary small positive value for a sufficiently large operator class. Some concrete examples of operators satisfying the conditions can be found in [45].

In order to represent a concrete example, consider the one-dimensional Schrodinger operator [45] (p. 194)

$$L=-d^2/d{x}^2+\varrho (x),\varrho (x)\ge a>0,\varrho \in CA{P}^{\infty }(\mathbb{R}).$$

The following asymptotical formula was obtained in the paper [48], see also [45] (p. 175)

$$n(r,L)=\sqrt{r}+r^{-{\displaystyle \frac{1}{2}}}\sum _{k=0}^{\infty }{d}_k{r}^{-k}.$$

Consider an operator with a discrete spectrum W satisfying the following conditions

$$\mathfrak{Re}W=L,|{\left(\mathfrak{Im}Wf,f\right)}_{L_2(\mathbb{R})}|\le 2^{-{\displaystyle \frac{1}{2}}}{\left(Lf,f\right)}_{L_2(\mathbb{R})}^q,f\in C_0^{\infty }(\mathbb{R}),{\parallel f\parallel }_{L_2(\mathbb{R})}=1,$$

(37)

$$0<q<1/2.$$

Applying Theorem 2.1 [15], i.e., implication (33), then using implication (34), we obtain

$$|{\mu }_n(W)|=n^2+o(n),n\in \mathbb{N},$$

hence $|{\mu }_{n+1}(W)|-|{\mu }_n(W)|>0$ for sufficiently large numbers $n\in \mathbb{N}.$ It follows that the operator $B:=W^{-1}$ has the sequence of the algebraic multiplicities of the lowest growth. The sectorial condition is satisfied by virtue of relation (37), i.e., $\Theta (B)\subset {\mathfrak{L}}_0(\theta ),\theta <\pi /4.$ Applying Lemma 1 [49] to the compact sectorial operator B we get

$$s_{2n-1}(B)\le \sqrt{2}sec\theta ·{\mu }_n(\mathfrak{Re}B),s_{2n}(B)\le \sqrt{2}sec\theta ·{\mu }_n(\mathfrak{Re}B),n\in \mathbb{N}.$$

Using the properties of the operator $L,$ condition (37), we can prove that the operator W satisfies conditions $\mathrm{H}1, \mathrm{H}2$ [29]. In accordance with the results [27] the conditions $\mathrm{H}1,\mathrm{H}2$ guarantee

$${\mu }_n^{-1}(L)\asymp {\mu }_n(\mathfrak{Re}B).$$

Analogously to (34), we get

$${\mu }_n(L)=n^2+o(n),n\in \mathbb{N}.$$

Combining these relations, we obtain the fact $B\in {\mathfrak{S}}_1.$ Thus, the operator B satisfies conditions of Theorem 1 and therefore operator W satisfies conditions of Theorem 2. This example illustrates efficiency of the obtained results in studying evolution equations containing a non-selfadjoint term.

Consider an operator class generated by a $C_0$ semigroup of contractions [29]. We can refer Theorem 5 [29] in accordance with which the operators belonging to the class satisfy conditions H1, H2 [29] and therefore have convenient, from the created theory point of view, properties such as a sectorial property, compactness of the resolvent, belonging to the Schatten-von Neumann class, etc. Consider the infinitesimal generator A of a $C_0$ semigroup of contractions, we can form the infinitesimal generator transform

$$Z_{\alpha }(A):=A^{\ast }GA+F{A}^{\alpha },\alpha \in [0,1),$$

where the symbols $G,F$ denote operators acting in $\mathfrak{H}.$ Taking into account Corollary 3.6 [50] (p. 11), Theorem 5 [29], we conclude that if $A^{-1}$ is compact, $F,G\in \mathcal{B}(\mathfrak{H}),$ G is strictly accretive, then $Z_{\alpha }(A)$ satisfies conditions H1–H2 [29]. Note that Theorem 5 [29] gives us a tool to describe spectral properties of the transform $Z_{\alpha }(A),$ in particular, we can establish the index of the Schatten-von Neumann class applying Theorem 3 [29]. Apparently, having known the index of the Schatten-von Neumann class, we can proceed to the next step applying results of this paragraph in order to verify fulfilment of Theorem 1 conditions.

4. Conclusions

In the paper, we have shown that the summation order in the Abel-Lidskii sense can be decreased to an arbitrary small positive value in the case corresponding to a sectorial operator belonging to the trace class under conditions imposed upon the growth of the algebraic multiplicities. In addition, we produce a number of fundamental propositions in the framework of functional analysis which may represent the interest themselves. The lemma on estimation of the characteristic determinant and the lemma on discharging of the spectrum create a prerequisite for further study. Thus, the invented technique forms a base for extension of the obtained results to an arbitrary Schatten-von Neumann class. Moreover, based upon the general scheme of reasonings and studying more detailed the issue on the brackets arrangement in the series, we can construct a qualitative theory describing the peculiarities of the series summation in the Abel-Lidkii sense. The application part appeals to the existence theorem for the abstract Cauchy problem for the fractional evolution equation that covers many concrete problems in the theory of differential equations. The results can be clearly illustrated on the operators having sufficiently slow growth of algebraic multiplicities. In its own turn, the latter property can be clearly expressed through the scale of spectral asymptotics for selfadjoint operators which is studied in detail in the final paragraph. In author’s opinion, the significant achievement of the paper is that the operator class generated by strictly continuous semigroups of contractions can be studied due to the obtained methods. The latter includes many well-known integro-differential operators such as the linear combination of the differential operator and the Kipriyanov fractional-differential operator, the linear combination of the differential operator and the Riesz potential, the perturbation of the difference operator. The author reasonably believes that the represented statements are principally novel in the framework of the abstract spectral theory while the obtained conclusions admit relevant applications.