Riemann-Liouville Operator in Weighted L_p Spaces via the Jacobi Series Expansion

In this paper we use the orthogonal system of the Jacobi polynomials as a tool to study the Riemann-Liouville fractional integral and derivative operators on a compact of the real axis.This approach has some advantages and allows us to complete the previously known results of the fractional calculus theory by means of reformulating them in a new quality. The proved theorem on the fractional integral operator action is formulated in terms of the Jacobi series coefficients and is of particular interest. We obtain a sufficient condition for a representation of a function by the fractional integral in terms of the Jacobi series coefficients. We consider several modifications of the Jacobi polynomials what gives us an opportunity to study the invariant property of the Riemann-Liouville operator. In this direction we have shown that the fractional integral operator, acting in the weighted spaces of Lebesgue square integrable functions, has a sequence of the included invariant subspaces.


Introduction
First, in this paper we aim to reformulate the well-known theorems on the Riemann-Liouville operator action in terms of the Jacobi series coefficients. In spite of that this type of problems was well studied by such mathematicians as Rubin B.S. [31], [32], [33], Vakulov B.G. [42], Samko S.G. [38], [39], Karapetyants N.K. [17], [18] (the results of [17], [31], [32] are also presented in [34]) in several spaces and for various generalizations of the fractional integral operator, the method suggested in this work allows us to notice interesting properties of the fractional integral and fractional derivative operators. We suggest using properties of the Jacobi polynomials for studying the Riemann-Liouville operator, but we should make a remark that this idea was previously used in the following papers [35], [4], [5], [10], [19], [36]. For instance: in the papers [5], [10] the operational matrices of the Riemann-Liouville fractional integral and the Caputo fractional derivative for shifted Jacobi polynomials were considered, in the paper [4] the fractional derivative formula was obtained applicably to the general class of polynomials introduced by Srivastava, in the paper [19] a general formulation for the fractional-order Legendre functions was constructed to obtain the solution of the fractional order differential equations. Also, which is interesting in itself, the fractional calculus theory was applied in [2], [37], [8] to study the Jacobi polynomials. However, our main interest lies in a rather different field of studying the mapping theorems for the Riemann-Liouville operator via the Jacobi polynomials. This approach gives us such an advantage as getting results in terms of the Jacobi series coefficients, let alone the concrete achievements. The central point of our method of studying is to use the basis property of the Jacobi polynomials system. In this way we aim to obtain a sufficient condition of existence and uniqueness of the Abel equation solution with the right part belonging to the weighted space of Lebesgue p-th integrable functions. Also, the usage of the weak topology gives us an opportunity to cover some cases in the mapping theorems that were not previously obtained. Besides, having filled some conditions gaps and formulated the unified result, we aim to systematize the mapping theorems established in the monograph [34].
Secondly, we notice that the question on existence of a non-trivial invariant subspace for an arbitrary linear operator acting in a Hilbert space is still relevant for today. In 1935 J. von Neumann proved that an arbitrary non-zero compact operator acting in a Hilbert space has a non-trivial invariant subspace [3]. This approach had got the further generalizations in the works [6], [13], but the established results are based on the compact property of the operator. In the general case the results [21], [24] are of particular interest. The overview of results in this direction can be found in [15], [9], [14]. Due to many difficulties in solving this problem in the general case, some scientists have paid attention to special cases and one of these cases was the Volterra integral operator acting in the space of Lebesgue squareintegrable functions on a compact of the real axis. The invariant subspaces of this operator were carefully studied and described in the papers [7], [11], [16]. We make an attempt to study invariant subspaces of the Riemann-Liouville fractional integral operator acting in the weighted space of Lebesgue square-integrable functions on a compact of the real axis. In this regard the following question is relevant: whether the Riemann-Liouville fractional integral has such an invariant subspace on which one would be selfadjoint.
The paper is organized as follows: In the second section the auxiliary formulas of fractional calculus are given as well as a brief remark on the Jacobi polynomials system basis property. In the third section the main results are presented, the mapping theorems established in the monograph [34] were systematized and reformulated in terms of the Jacoby series coefficients, the invariant subspaces of the Riemann-Liouville operator were studied. The conclusions are given in the fourth section.

Some fractional calculus formulas
Throughout this paper we consider complex functions of a real variable, we use the following denotation for weighted complex Lebesgue spaces L p (I, ω), 1 ≤ p < ∞, where I = (a, b) is an interval of the real axis and the weighted function ω is a real-valued function. Also we use the denotation p ′ = p/(p − 1). If ω = 1, then we use the notation L p (I). Using the denotations of the paper [34], let us define the left-side, right-side fractional integrals and derivatives of real order respectively where I α a+ (L 1 ), I α b− (L 1 ) are the classes of functions which can be represented by the fractional integrals (see [34, p.43]). Further, we use as a domain of definition of the fractional differential operators mainly the set of polynomials on which these operators are well defined. We use the shorthand notation L 2 := L 2 (I) and denote by (·, ·) the inner product on the Hilbert space L 2 (I). Using Definition 1.5 [34, p.4] we consider Denote by C, C i , i ∈ N positive real constants. We mean that the values of C can be different in various parts of formulas, but the values of C i , i ∈ N are certain. We use the following special denotation Further, we need the following formulas for multiple integrals. Note that under the assumption ϕ ∈ L 1 (I), we have Suppose f (x) ∈ AC n (Ī), n ∈ N; then using the previous formulas we have the representations in the right-side case 2.2 Riemann-Liouville operator via the Jacobi polynomials The orthonormal system of the Jacobi polynomials is denoted by where the normalized multiplier δ n (β, γ) is defined by the formula the orthogonal polynomials y (β,γ) n are defined by the formula For convenience, we use the following functions If misunderstanding does not appear, we will use the shorthand denotations in various parts of this work In such cases we would like reader see carefully the denotations corresponding to a concrete paragraph. Specifically, in the case of the Jacobi polynomials, when β = γ = 0, we have the Legendre polynomials. If we consider the Hilbert space L 2 (I), then the Legendre orthonormal system has a basis property due to the general property of complete orthonormal systems in Hilbert spaces, but the question on the basis property of the Legendre system for an arbitrary p ≥ 1, p = 2 had been still relevant until half of the last century. In the direction of solving this problem the following works are known [26], [28], [29], [30]. In particular, in the paper [28] Pollard H. proved that the Legendre system has a basis property in the case 4/3 < p < 4 and for the values of p ∈ [1, 4/3] ∪ [4, ∞) the Legendre system does not have a basis property in L p (I) space. The cases p = 4/3, p = 4 were considered by Newman J. and Rudin W. in the paper [26] where it is proved that in these cases the Legendre system also does not have a basis property in L p (I) space. It is worth noting that the criterion of a basis property for the Jacobi polynomials was proved by Pollard H. in the work [30]. In that paper Pollard H. formulated the theorem proposing that the Jacobi polynomials have a basic property in the space L p (I 0 , ω), I 0 := (−1, 1), β, γ ≥ −1/2, M (β, γ) < p < m(β, γ) and do not have a basis property, when p < M (β, γ) or p > m(β, γ), where m(β, γ) = 4 min β + 1 2β + 1 , γ + 1 2γ + 1 , M (β, γ) = 4 max β + 1 2β + 3 , γ + 1 2γ + 3 .
However, this result was subsequently improved by Muckenhoupt B. in the paper [23]. Note that the linear transform l : shows us that all results of the orthonormal polynomials theory obtained for the segment [−1, 1] are true for the segment [a, b] ⊂ R. We use the denotation S k f := k n=0 f n p (β,γ) n , k ∈ N 0 , where f n are the Jacobi series coefficients of the function f. Consider the orthonormal Jacobi polynomials Further, we need some formulas. Using the Leibnitz formula, we get Using again the Leibnitz formula, we obtain where c = max {0 , k + i − n} , k ≤ n . In accordance with (5), we have In the same way, we get Hence n (γ, β). Using the Taylor series expansion for the Jacobi polynomials, we get Applying the formulas (2.44),(2.45) of the fractional integral and derivative of a power function [34, p.40], we obtain here we used the formal denotation I −α a+ := D α a+ . Thus, using integration by parts, we get In the same way, we get

Using the denotation
We claim the following formulas without any proof because of the absolute analogy with the proof corresponding to the fractional integral operators Further, we use the following denotations This allows us to consider the integro-differential operators in the matrix form of notation. Throughout this paper the results are formulated and proved for the left-side case. One may reformulate them for the right-side case with no difficulty.

Mapping theorems
The following lemma aims to establish more simplified and at the same time applicable form of the results proven in Theorem 3.10 [34, p.78], Theorem 3.12 [34, p.81] and is devoted to the description of the operator I α a+ action in the space L p (I, ω). More precisely, these theorems describe the action I α a+ : L p (I, ω) → L q (I, r) with rather inconveniently formulated conditions, from the point of view of operator theory, regarding to the weighted functions and indexes p, q. To justify this claim, we can easily see that there are some cases in the theorems conditions for which the bounded action I α a+ : L p (I, ω) → L p (I, ω), α ∈ (0, 1), ω(x) = (x − a) β (b − x) γ does not follow easily from the theorems, for instance in the case 2 < p < 1/(1 − α), β ∈ R, 0 < γ ≤ αp − 1, the mentioned above bounded action of I α a+ cannot be obtained by using the theorems and estimating, as we shall see further the proof of this fact requires to involve the weak topology methods.
Proof. By direct calculation, we can verify that β satisfies the inequality 2t 2 + t − 1 < 0. We see that Let us substitute β for t, we have Hence β < p − 1. We have absolutely analogous reasoning for γ i.e. γ < p − 1. Let us consider the various relations between p and α.
It is easily shown that Solving the quadratic equality we can verify that under the assumptions .
It is obvious that γ(1 − p ′ ) < p ′ − 1. Combining relation (14) and Theorem 3.8 [34, p.74], we obtain Taking into account the above considerations, we obtain Thus, we get m ∈ L p ′ (I, ω). Using the Hölder inequality and the previous reasoning, we get Hence in accordance with the consequence of the Fubini theorem, we get Consider the functional .
Applying (15), we obtain We see that the previous inequality is true for all linear combinations Since it can easily be checked that M (β, γ) < p ′ < m(β, γ), then in accordance with the results of the paper [30] the system {p m } ∞ 0 has a basis property in the space L p ′ (I, ω). Using this fact, we pass to the limit in both sides of inequality (18), thus we get In the terms of the given above denotation we can write | I α a+ f, g L 2 (I, ω) | ≤ C f Lp(I, ω) g L p ′ (I, ω) , ∀g ∈ L p ′ (I, ω) .
In its turn, this inequality can be rewritten in the following form Hence the set is weekly bounded. Therefore, in accordance with the well-known theorem this set is bounded with respect to the norm L p (I, ω). It implies that (12) holds. If β < 0, then it is easy to show that .
The results of the monograph [34] (see Chapter 1) give us a description of the fractional integral mapping properties in the space L p (I, ω), 1 < p < ∞, p = 1/α, where ω is some power function. Actually, the following question is still relevant. What does happen in the case p = 1/α ? In the nonweighted case, the approach to this question is given in the paper [27]. Also, it can be found in a more convenient form in the monograph [34, p.92], there the following inequality is given It is remarkable that there is no mention on the weighted case in the historical review of the monograph [34]. In contrast to the said above approaches, we obtain a description of the fractional integral mapping properties in the space L p (I, ω) in terms of the Jacobi series coefficients. This approach is principally different from ones used in [34], in particular it allows us to avoid problems confected with the case p = 1/α. Further, in this section we deal with the normalized Jacobi polynomials p then

This theorem can be formulated in the matrix form
Proof. Note, that according to the results of the paper [30] the system of the normalized Jacobi polynomials has a basis property in L p (I, ω), M (β, γ) < p < m(β, γ). Hence Applying first formula (9), we obtain f m = I α a+ ψ, p m Lp(I, ω) = ∞ n=0 (−1) n ψ n A α,β,γ mn .
Using denotations (11), we obtain the matrix form for the statement of this theorem.
The following result is formulated in terms of the Jacobi series coefficients and is devoted to the representation of a function by the fractional integral. Consider the Abel equation under most general assumptions relative to the right part If the next conditions hold then there exists a unique solution of equation (21) in the class L 1 (I) (see Theorem 2.1 [34, p.31]). The sufficient conditions for existence and uniqueness of the Abel equation solution are established in the following theorem under the minimum assumptions relative to the right part of (21). In comparison with the ordinary Abel equation, we avoid imposing conditions similar to (22), moreover we refuse the assumption that the right part is a Lebesgue integrable function.
, M (β, γ) < p < m(β, γ), the right part of equation (21) such that then there exists a unique solution of equation (21) in L p (I, ω), the solution belongs to L q (I, ω), where: q = p, when 0 ≤ λ ≤ 1/2 ; q = max{p, t}, t < (2s − 1)/(s − λ), when 1/2 < λ < s (s = 3/2 + max{β, γ}); q is arbitrary large, when λ ≥ s. Moreover the solution is represented by a convergent in L q (I, ω) series Proof. Applying first formula (10), we obtain the following relation We can easily verify that M (β, γ) < p ′ < m(β, γ). Hence due to Theorem A [30] the system {p n } ∞ 0 has a basis property in the space L p ′ (I, ω). Since relation (25) holds and the sequence D α a+ S k f ∞ 0 is bounded in the sense of norm L p (I, ω), then due to the well-known theorem, we have that the sequence D α a+ S k f ∞ 0 converges weakly to some function ψ ∈ L p (I, ω Let us show that ω −1 I α b− ϕ m ∈ L p ′ (I, ω). For this purpose consider the functional Using the Hölder inequality, Lemma 2.1, we have Hence using the Dirichlet formula, we have By virtue of this fact, we can rewrite relation (26) in the following form Using the Riesz representation theorem, we obtain ω −1 I α b− ϕ m ∈ L p ′ (I, ω). Hence, we get Using the Dirichlet formula, we obtain Hence (S k f, p m ) L 2 (I, ω) −→ I α a+ ψ, p m L 2 (I, ω) , k → ∞, m ∈ N 0 .
Taking into account that we obtain I α a+ ψ, p m L 2 (I, ω) = f m , m ∈ N 0 . Using the uniqueness property of the Jacobi series expansion, we obtain I α a+ ψ = f almost everywhere. Hence there exists a solution of the Abel equation (21). If we assume that there exists another solution φ ∈ L p (I, ω), then we get I α a+ ψ = I α a+ φ almost everywhere. Consider the function η ∈ C ∞ 0 (I). Using Theorem 2.4 [34, p.44] and the Dirichlet formula, we have Consider an interval I ′ ⊂ I. Note that ψ, φ ∈ L p (I ′ ), ∀I ′ . Since C ∞ 0 (I ′ ) ⊂ C ∞ 0 (I), here we assume that the functions belonging to C ∞ 0 (I ′ ) have the zero extension outside of I ′ , then we obtain We claim that ψ = φ. Hence in accordance with the consequence of the Hahn-Banach theorem there exists the element ϑ ∈ L p ′ (I ′ ), such that On the other hand, there exists the sequence Thus we come to contradiction. Hence ψ = φ almost everywhere on I ′ , ∀I ′ ⊂ I. It implies that ψ = φ almost everywhere on I. The uniqueness has been proved. Now let us proceed to the following part of the proof. Note that it was proved above ψ ∈ L p (I, ω), when 0 ≤ λ < ∞. Let us show that ψ ∈ L q (I, ω), where q < (2s − 1)/(s − λ), 1/2 < λ < s. In accordance with the reasoning given above, we have Combining this fact with (25), we get Using the theorem conditions, we have Now we need an adopted version of the Zigmund-Marczinkevich theorem (see [22]), which establishes the following. Let {φ n } be an orthoghonal system on the segmentĪ and φ n L∞(I) ≤ M n , (n = 1, 2, ...), where M n is a monotone increasing sequence of real numbers. If q ≥ 2 and we have then the series ∞ n=1 c n φ n (x) converges in L q (I) to some function f ∈ L q (I) and f Lq(I) ≤ CΩ q (c). We aim to apply this theorem to the case of the Jacobi system, however we need some auxiliary reasoning. As the matter of fact, we deal with the weighted L p (I, ω) spaces, but the Zigmund-Marczinkevich theorem in its pure form formulated in terms of the non-weighted case. Consider the following change of the variable x a ω(t)dt = τ. For the solution ψ ∈ L p (I, ω), we have whereψ(τ ) = ψ(κ(τ )), φ n (τ ) = p n (κ(τ )), κ(τ ) = (b−a) −(β+γ+1) B −1 τ (β+1, γ+1), B = (b−a) β+γ+1 B(β+ 1, γ + 1). Hence, if we note the estimate |p n (x)| ≤ Cn a+1/2 , a = max{β, γ}, x ∈Ī (see Theorem 7.3 [40, p.288]), then due to the change of the variable, we have |φ n (τ )| ≤ V n , τ ∈ [0, B], V n = Cn a+1/2 . Also, it is clear that (φ m , φ n ) L 2 (0,B) = δ mn , where δ mn is the Kronecker symbol. Thus {φ m } ∞ 0 is the orthonormal system on [0, B] that satisfies the conditions of the Zigmund-Marczinkevich theorem. It can easily be checked that due to the theorem conditions the following series is convergent For the values λ ≥ s, series (31) converges for an arbitrary positive q. In accordance with given above, Thus all conditions of the Zigmund-Marczinkevich theorem are fulfilled. Hence, we can conclude that there exists a function ν such that the next estimate holds Since the system {p m } ∞ 0 has a basis property in L p (I, ω), then it is not hard to prove that the system {φ m } ∞ 0 has a basis property in L p (0, B). Since the functions ν andψ have the same Jacobi series coefficients, then ν =ψ almost everywhere on (0, B). By virtue of the chosen change of the variable, we obtain ψ Lq (I,ω) = ψ Lq(0,B) . Consequently, the solution ψ belongs to the space L q (I, ω), q < (2s − 1)/(s − λ), when 1/2 < λ < s and the index q is arbitrary large, when λ ≥ s. Taking into account (30) and applying the Zigmund-Marczinkevich theorem, we have Using the inverse change of the variable and applying (28), we obtain (24).

Non-simple property problem
The questions related to existence of such an invariant subspace of the operator that the operator restriction to the subspace is selfadjont (the so-called non-simple property [12, p.275] ) are still relevant for today. Thanks to the powerful tool provided by the Jacobi polynomials theory, we are able to approach a little close to solving this problem for the Riemann-Liouville operator.
In this section we deal with the so-called normalized ultraspherical polynomials p (β, β) n (x) in the weighted space L p (I, ω), ω(x) = [(x − a)(b − x)] β , β ≥ −1/2, 1 ≤ p < ∞. In accordance with [29] the system of the normalized ultraspherical polynomials has a basis property in L p (I, ω), if 1 − 1/(3 + 2β) < p/2 < 1 + 1/(1 + 2β), λ = β + 1/2 and does not have a basis property, if 1/2 ≤ p/2 < 1 − 1/(3 + 2β) or p/2 > 1 + 1/(1 + 2β). Having noticed that A α,β,β mn = A α,β,β nm , m, n ∈ N 0 , using formulas (9), we obtain b a I α Taking into account these formulas we conclude that the fractional integral operator is symmetric in the subspaces of L 2 (I, ω) generated respectively by the even system {p 2k } ∞ 0 and the odd system {p 2k+1 } ∞ 0 of the normalized ultraspherical polynomials. Let us denote by L + 2 (I, ω), L − 2 (I, ω) these subspaces respectively. The following theorem gives us an alternative. Proof. We provide the proof only for the left-side case, since the proof corresponding to the right case is analogous and can be obtained by simple repetition. Let us show that the operator I α a+ : L 2 (I, ω) → L 2 (I, ω) is compact. Using Theorem 3.12 [34, p.81], we have the estimate where It can easily be checked that in the case (β > 2α − 1), we have and in the case (β ≤ 2α − 1), we have Thus, using estimate (34), we obtain Now let us use the Kolmogorov criterion of compactness (see [20]), which claims that a set in the space L p (I, ω), 1 ≤ p < ∞ is compact, if this set is bounded and equicontinuous with respect to the norm L p (I, ω). Note that by virtue of (35) the set I α a+ (N) is bounded in L 2 (I, ω), where N := f : f L 2 (I, ω) ≤ M, M > 0 . Using (34), we get I α a+ f H λ 0 (Ī, r) ≤ C 1 , ∀f ∈ N. Hence in accordance with the definition, we have |(I α , where t is a sufficiently small positive number. Under the assumption that functions have a zero extension outside ofĪ, we have Assume that f ∈ N and consider the case (β ≤ 2α − 1). Note that due to Theorem 3.12 [34, p.81], we obtainĨ Using the Minkowski inequality, we get As before, applying Theorem 3.12 [34, p.81], we get I 1 ≤ Ct α−1/2 . Using the inequality (τ + 1) ν < τ ν + 1, τ > 1, 0 < ν < 1, we obtain In the same way, using the inequality (τ − 1) ν > τ ν − 1, τ > 1, 0 < ν < 1, we get Since r(x) is a product of the functions that satisfy the Hölder condition, then it is not hard to prove that Using the fact I α a+ f H λ 0 (Ī,r ≤ C 1 , we get Taking into account the above reasoning, we have ≤ Ct 2β−µ ; Hence we conclude that I 2 ≤ Ct δ 1 and as a consequence, we obtain I ≤ Ct δ 2 , where δ 1 , δ 2 are some positive numbers. To achieve the case (β > 2α − 1) we should just repeat the previous reasoning having replaced µ by β. The proof is omitted. Thus in both cases considered above we obtain It implies that the conditions of the Kolmogorov criterion of compactness [20] are fulfilled. Hence any bounded set with respect to the norm L 2 (I, ω) has a compact image. Therefore the operator I α a+ : L 2 (I, ω) → L 2 (I, ω) is compact. Now applying the von Neumann theorem [1, p.204], we conclude that there exists a non-trivial invariant subspace of the operator I α a+ , which we denote by M. On the other hand, using the basis property of the system {p n } ∞ 0 , we have L 2 (I, ω) = L + 2 (I, ω) ⊕ L − 2 (I, ω). It is quite sensible to assume that M ∩ L + 2 (I, ω) = 0, M ∩ L − 2 (I, ω) = 0. If we assume otherwise, then we have an invariant subspace on which the operator I α a+ , by virtue of formulas (3.2), is selfadjoin and we get the first statement of the alternative. Continuing this line of reasoning, we see that under the assumption excluding the first statement of the alternative we come to conclusion that this process can be finished only in the case, when on some step we get a finite-dimensional invariant subspace. We claim that it can not be! The proof is by reductio ad absurdum. Assume the converse, then we obtain a finite-dimensional restrictionĨ α a+ of the operator I α a+ . Applying the reasoning of Theorem 2, we can easily prove that the point zero is not an eigenvalue of the operator I α a+ , hence one is not an eigenvalue of the operatorĨ α a+ . It implies that in accordance with the fundamental theorem of algebra the operatorĨ α a+ has at least one non-zero eigenvalue (sinceĨ α a+ is finite-dimensional). It is clear that this eigenvalue is an eigenvalue of the operator I α a+ . We can write Further, we use the method described in [41, p.14]. Using the Cauchy Schwarz inequality, we get where (37), we get Continuing this process, we obtain ..
Since |λ| −2(n+1) J n /n! → 0, n → ∞, then f (x) = 0, x ∈ I. We have obtained the contradiction with (36), which allows us to conclude that there does not exist a finite dimensional invariant subspace. It implies that we have the sequence of the included invariant subspaces

Conclusions
In this paper, the first our aim is to reformulate in terms of the Jacoby series coefficients the previously known theorems describing the Riemann-Liouville operator action in the weighted spaces of Lebesgue p-th power integrable functions, the second aim is to approach a little bit closer to solving the problem: whether the Riemann-Liouville operator acting in the weighted space of Lebesgue square integrable functions is simple. The approach, which was used in the paper is in the following: to use the Jacobi polynomials special properties that alow us to apply novel methods of functional analysis and theory of functions of a real variable, which are rather different in comparison with the perviously applied methods for studying the Riemann-Liouville operator. Besides the main results of the paper, we stress that there was arranged some systematization of the previously known facts of the Riemann-Liouville operator action in the weighted spaces of Lebesgue p-th power integrable functions, when the weighted function is represented by some kind of a power function. It should be noted that the previously known description of the Riemann-Liouville operator action in the weighted spaces of Lebesgue p-th power integrable functions consists of some theorems in which the conditions imposed on the weight function have the gaps, i.e. some cases corresponding to the concrete weighted functions was not considered. Motivated by this, among the unification of the known results, we managed to fill the gaps of the conditions and formulated this result as a separate lemma. The following main results were obtained in terms of the Jacoby series coefficients: the theorem on the Riemann-Liouville operator direct action was proved, the existence and uniqueness theorem for Abel equation in the weighted spaces of Lebesgue p-th power integrable functions was proved and the solution formula was given, the alternative in accordance with which the Riemann-Liouville operator is either simple or one has the sequence of the included invariant subspaces was established. Note that these results give us such a view of the fractional calculus that has a lot of advantages. For instance, we can reformulate Theorem 2 under more general assumptions relative to the integral operator on the left side of equation (21), at the same time having preserved the main scheme of the reasonings. In this case the most important problem may be, in what way we are able to calculate the coefficients given by formula (10). Besides, the notorious case p = 1/α, which was successfully achieved in this paper is also worth noticing. Thus the obtained results make a prerequisite of researching in such a direction of fractional calculus.