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Abstract evolution equations with an operator function at the right-hand side

In this paper, we deal with non-selfadjoint operators with the compact resolvent. Having inspired by the Lidskii idea involving a notion of convergence of a series on the root vectors of the operator in a weaker – Abel-Lidskii sense, we proceed constructing theory in the direction. The main concept of the paper is a generalization of the spectral theorem for a non-selfadjoint operator. In this way, we come to the deﬁnition of the function of an unbounded non-selfadjoint operator. As an application, we notice some approaches allowing us to principally broaden conditions imposed on the right-hand side of the evolution equation in the abstract Hilbert space and produce some concrete examples signiﬁcant in applied sciences. In conection with this such operator as a Riemann-Liouville fractional diﬀerential operator, Kipriyanov fractional diﬀerential operator, Riesz potential are involved.


Introduction
We should recall that the concept of the spectral theorem for a selfadgoint operator is based on the notion of a spectral family or the decomposition of the identical operator. Constructing a spectral family, we can define the selfadjoint operator using the Riemann integral, it is the very statement of the spectral theorem for a selfadjoint operator. Using the same scheme, we come to a notion of the operator function of a selfadjoint operator. The idea can be clearly demonstrated if we consider the well-known representation of the compact selfadjoint operator as a series on its eigenvectors. The case corresponding to the non-selfadjoint operator is not so clear but we can adopt some notions and techniques to obtain familiar results. Firstly, we should note that the question regarding decompositions of the operator on the series of eigenvectors (root vectors) is rather complicated and deserves to be considered itself. For this purpose, we need to involve some generalized notions of the series convergence, we are compelled to understand it in one or another sense, we mean Bari, Riesz, Abel (Abel-Lidskii) senses of the series convergence [1], [2]. A reasonable question that appears is about minimal conditions that guaranty the desired result, for instance in the mentioned papers there considered a domain of the parabolic type containing the spectrum of the operator. In the paper [1], non-salfadjoint operators with the special condition imposed on the numerical range of values are considered. The main advantage of this result is a weak condition imposed upon the numerical range of values comparatively with the sectorial condition (see definition of the sectorial operator). Thus, the convergence in the Abel-Lidskii sense was established for an operator class wider than the class of sectorial operators. Here, we make a comparison between results devoted to operators with the discrete spectra and operators with the compact resolvent, for they can be easily reformulated from one to another realm. The central idea of this paper is to represent propositions analogous to the spectral theorem, the main obstacle that appears is how to define an analogue of a spectral family or decomposition of the identical operator. Thus, one of the paper challenges is to find a harmonious way of reformulating the main principles of the spectral theorem, taking into account the peculiarities of the convergence in the mentioned above senses. Here, we should make a brief digression and consider a theoretical background that allows us to obtain such exotic results.
We begin with the paper [10] which major contribution to the theory is a sufficient conditions of the Abel-Lidskii basis property of the root functions system for a sectorial non-selfadjoint operator of the special type. Considering such an operator class we strengthen a little the condition regarding the semi-angle of the sector, but weaken a great deal conditions regarding the involved parameters. Moreover, the central aim generates some prerequisites to consider technical peculiarities such as a newly constructed sequence of contours of the power type on the contrary to the Lidskii results [15], where a sequence of the contours of the exponential type was considered. We may say that a clarification of the results [15] devoted to the decomposition on the root vector system of the non-selfadjoint operator was obtained. In that study, we used a technique of the entire function theory and introduced a so-called Schatten-von Neumann class of the convergence exponent. Using a sequence of contours of the power type, we invented a peculiar method how to calculate a contour integral, involved in the problem in its general statement, for strictly accretive operators satisfying special conditions formulated in terms of the norm.
Note that the application part of the direction appeals to the theory of differential equations. In particular, the existence and uniqueness theorems for evolution equations with the right-hand side -a differential operator with a fractional derivative in final terms are covered by the invented abstract method. In this regard such operator as a Riemann-Liouville fractional differential operator, Kipriyanov operator, Riesz potential, difference operator are involved. Note that analysis of the required conditions imposed upon the right-hand side of the evolution equations that are in the scope leads us to relevance of the central idea of the paper [10].We ought to note a well-known fact [25], [7] that a particular interest appears in the case when a senior term of the operator is not selfadjoint, for in the contrary case there is a plenty of results devoted to the topic wherein the following papers are well-known [4], [13], [16], [17], [25]. 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 ore normal operator. In other cases, the methods of the papers [8], [7] become relevant and allow us to study spectral properties of operators whether we have the mentioned above representation or not. Here, we should remark that the methods [7] can be used in the natural way, if we deal with abstract constructions formulated in terms of the semigroup theory [9]. The central challenge of the latter paper is how to create a model representing a composition of fractional differential operators in terms of the semigroup theory. We should note that motivation arises in connection with the fact that a second order differential operator can be represented as a some kind of a transform of the infinitesimal generator of a shift semigroup. Having been inspired by novelty of the idea, we generalize a differential operator with a fractional integro-differential composition in the final terms to some transform of the corresponding infinitesimal generator of the shift semigroup. Having applied the methods [7], we managed to study spectral properties of the infinitesimal generator transform and obtain an outstanding result -asymptotic equivalence between the real component of the resolvent and the resolvent of the real component of the operator. The relevance is based on the fact that the asymptotic formula for the operator real component in most cases can be established due to well-known results [23].
In its own turn, the application of results connected with the basis property [10] covers many problems in the framework of the theory of evolution equations [11]. The peculiar contribution of this paper is the obtained formula for the solution of the evolution equation with the special conditions imposed upon the right-hand side, where the derivative at the left-hand side is supposed to be of the fractional order. Here, we should remind that involving the notion of the operator function, we broaden a great deal a class corresponding to the right-hand side. It gives us a prerequisite to consider that the offered approach is relevant.

Preliminaries
Let C, C i , i ∈ N 0 be real constants. We assume that a value of C is positive and can be different  [2] the dimension of the root vectors subspace corresponding to a certain eigenvalue λ k is called the algebraic multiplicity of the eigenvalue λ k . Denote by n(r) a function equals to the number of the elements of the sequence {a n } ∞ 1 , |a n | ↑ ∞ within the circle |z| < r. Let A be a compact operator, denote by n A (r) counting function a function n(r) corresponding to the sequence {s −1 i (A)} ∞ 1 . Let S p (H), 0 < p < ∞ be a Schatten-von Neumann class and S ∞ (H) be the set of compact operators. Suppose L is an operator with a compact resolvent and s n (R L ) ≤ C n −µ , n ∈ N, 0 ≤ µ < ∞; then we denote by µ(L) order of the operator L in accordance with the definition given in the paper [25]. Denote by ReL := (L + L * ) /2, ImL := (L − L * ) /2i the real and imaginary components of an operator L respectively. In accordance with the terminology of the monograph [3] the set Θ(L) := {z ∈ C : z = (Lf, f ) H , f ∈ D(L), f H = 1} is called the numerical range of an operator L. An operator L is called sectorial if its numerical range belongs to a closed sector L ι (θ) := {ζ : | arg(ζ − ι)| ≤ θ < π/2}, where ι is the vertex and θ is the semi-angle of the sector L ι (θ). If we want to stress the correspondence between ι and θ, then we will write θ ι . By the convergence exponent ρ of the sequence {a n } ∞ 1 ⊂ C, a n = 0, a n → ∞ we mean the greatest lower bound for such numbers λ that the following series converges More detailed information can be found in [14]. Denote byS ρ (H) the class of the operators such thatS ρ (H) := {T ∈ S ρ+ε , T ∈ S ρ−ε , ∀ε > 0}, we will call it Schatten-von Neumann class of the convergence exponent. Everywhere further, unless otherwise stated, we use notations of the papers [2], [3], [5], [6], [24].

Convergence in the Abel-Lidsky sense
In this subsection, we reformulate results obtained by Lidskii [15] in a more convenient form applicable to the reasonings of this paper. However, let us begin our narrative. In accordance with the Hilbert theorem (see [22], [2, p.32]) 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 where both summands are invariant subspaces regarding the operator B, the first one is a finite dimensional root subspace corresponding to the eigenvalue µ q and the second one is a subspace wherein the operator B − µ q I is invertible. Let n q is a dimension of N q and let B q is the operator induced in N q . We can choose a basis (Jordan basis) in N q that consists of Jordan chains of eigenvectors and root vectors of the operator B q . Each chain e q ξ , e q ξ +1 , ..., e q ξ +k , k ∈ N 0 , where e q ξ , ξ = 1, 2, ..., m are the eigenvectors corresponding to the eigenvalue µ q and other terms are root vectors, can be transformed by the operator B in accordance with the following formulas Be q ξ = µ q e q ξ , Be q ξ +1 = µ q e q ξ +1 + e q ξ , ..., Be q ξ +k = µ q e q ξ +k + e q ξ +k−1 .
Considering the sequence {µ q } ∞ 1 of the eigenvalues of the operator B and choosing a Jordan basis in each corresponding space N q , we can arrange a system of vectors {e i } ∞ 1 which we will call a system of the root vectors or following Lidskii a system of the major vectors of the operator B.
Assume that e 1 , e 2 , ..., e nq is the Jordan basis in the subspace N q . We can prove easily (see [15, p.14]) that there exists a corresponding biorthogonal basis g 1 , g 2 , ..., g nq in the subspace M ⊥ q . Using the reasonings [10], we conclude that {g i } nq 1 consists of the Jordan chains of the operator B * which correspond to the Jordan chains (2) due to the following formula It is not hard to prove that the set {g ν } n j 1 , j = i is orthogonal to the set {e ν } n i 1 (see [10]). Gathering the sets {g ν } n j 1 , j = 1, 2, ..., we can obviously create a biorthogonal system {g n } ∞ 1 with respect to the system of the major vectors of the operator B. It is rather reasonable to call it as a system of the major vectors of the operator B * . Note that if an element f ∈ H allows a decomposition in the strong sense f = ∞ n=1 e n c n , c n ∈ C, then by virtue of the biorthogonal system existing, we can claim that such a representation is unique. Further, let us come to the previously made agrement that the vectors in each Jordan chain are arranged in the same order as in (2) i.e. at the first place there stands an eigenvector. It is clear that under such an assumption we have where k(q ξ ) + 1 is a number of elements in the q ξ -th Jourdan chain. In particular, if the vector e q ξ is included to the major system solo, there does not exist a root vector corresponding to the same eigenvalue, then Note that in accordance with the property of the biorthogonal sequences, we can expect that the denominators equal to one in the previous two relations. Consider a formal series corresponding to a decomposition on the major vectors of the operator B where each number n corresponds to a number q ξ + i (thus, the coefficients c n are defined in accordance with the above and numerated in a simplest way). Consider a set of functions with respect to a real parameter t Here we should note that if ϕ = const, then we have a set of polynomials, what is in the origin of the concept, see [15]. Consider a series where the coefficients c n (t) are defined in accordance with the correspondence between the indexes n and q ξ + i in the following way here λ q = 1/µ q is a characteristic number corresponding to e q ξ . It is clear that in any case, we have a limit c n (t) →c n , t → + 0, where a valuec n can be calculated directly due to the formula (4). For instance in the case ϕ = 1, we havec n = c n . Generalizing the definition given in [15, p.17], we will say that series (3) converges to the element f in the sense (B, ϕ, 1), if there exists a sequence of the natural numbers {N j } ∞ 1 such that Note that sums of the latter relation forms a subsequence of the partial sums of the series (3). We need the following lemmas [15], in the adopted form, also see [10]. Further, considering an arbitrary compact operator B : H → H such that Θ(B) ⊂ L 0 (θ), θ < π, we put the following contour in correspondence to the operator where ε > 0 is an arbitrary small number, the number r is chosen so that the operator (I − λB) −1 is regular within the corresponding closed circle. Here, we should note that the compactness property of B gives us the fact (I − λB) −1 ∈ B(H), λ ∈ C \ intγ (B). It can be proved easily if we note that in accordance with the Corollary 3.3 [3, p.268], we have P(B) ⊂ C \ Θ(B).

Lemma 2.
Assume that a compact operator B satisfies the condition B ∈S ρ , then for arbitrary numbers R, κ such that R > 0, 0 < κ < 1, there exists a circle |λ| =R, (1 − κ)R <R < R, so that the following estimate holds

Main results
In this section, we consider an adopted technique allowing us to reformulate results [10] in accordance with our challenge to make a harmonious generalization of a notion -a function of an unbounded non-selfadjoint operator. Firstly, we consider statements with the necessary refinement caused by the involved functions, here we should note that a particular case corresponding to a constant function was considered by Lidskii [15]. The interesting fact is that involving a non-constant function we are still able to consider a patrial sums of the series on the root vectors corresponding to some groups of eigenvalues. Secondly, we find conditions that guarantee convergence of the involved integral construction and formulate a main theorem giving us a tool for further study. We consider an analogue of the spectral family having involved the operators similar to Riesz projectors (see [2, p.20] ) and using a notion of the convergence in the Abel-Lidskii sense. Finally, we discus an approach that we can realize for applying the abstract theoretical results to the evolution equations.
where m(q) is a geometrical multiplicity of the q-th eigenvalue, k(q ξ ) + 1 is a number of elements in the q ξ -th Jourdan chain, Proof. Consider an integral where the interior of the contour γ q does not contain any poles of the operator (I − λB) −1 , except of λ q . Assume that N q is a root space corresponding to λ q and consider a Jordan basis {e q ξ +i }, i = 0, 1, ..., k(q ξ ), ξ = 1, 2, ..., m(q) in N q . Using decomposition of the Hilbert space in the direct sum (1), we can represent an element Note that the operator function ϕ(λ)e −λt B(I − λB) −1 f 2 is regular in the interior of the contour γ q , it follows from the fact that λ q ia a normal eigenvalue (see the supplementary information). Hence, we have Using the formula we obtain Now, let us decompose the element f 1 on the corresponding Jordan basis, we have In accordance with the relation (2), we get Be q ξ = µ q e q ξ , Be q ξ +1 = µ q e q ξ +1 + e q ξ , ..., Be q ξ +k = µ q e q ξ +k + e q ξ +k−1 .
Using this formula, we can prove the following relation Note that the case i = 0 is trivial. Consider a case, when i > 0, we have Using these formulas, we obtain what gives us the desired result. Now, substituting (5),(6), we get Note that the function ϕ(1/ζ) is analytic inside the interior ofγ q , hence 1 2πi Changing the indexes, we have where c q ξ +j (t) := e −λqt k(q ξ )−j m=0 c q ξ +j+m Q m (ϕ, λ q , t).
The proof is complete.
Note that using the reasonings of the last lemma, it is not hard to prove that Bellow, we consider an invertible operator B and use a notation W := B −1 . Consider a function ϕ that can be represented by a Laurent series about the point zero. Denote by a formal construction called by a function of the operator, where c n are coefficients corresponding to the function ϕ. The lemmas given bellow are devoted to the study of the conditions under which being imposed the series (7) converges on some elements of the Hilbert space H, thus the operator ϕ(W ) is defined. We will also consider a closureφ(W ) of the operator ϕ(W ).
Lemma 5. Assume that the operator B satisfies conditions of Lemma 1, Proof. Consider a decomposition of the Laurent series on two terms Consider an obvious relation It gives us the following representation 1 2πi Let us show that I 2 (t) = 0, define a contour γ R (B) := Fr {int γ(B) ∩ {λ : r < |λ| < R}} and let us prove that Consider a decomposition of the contour γ R (B) on termsγ R := {λ : |λ| = R, |argλ| ≤ θ + ε}, γ R + := {λ : r ≤ |λ| < R, argλ = θ + ε}, γ R − := {λ : r < |λ| < R, argλ = −θ − ε}. It is clear that 1 2πi Having noticed that I R (t) = 0, since the operator function under the integral is analytic inside the countour, we come to the conclusion that to obtain the desired result, we should show We have Using the condition |argλ| < π/2, we have Reλ ≥ |λ| cos(π/2 − δ) = |λ| sin δ, where δ is a sufficiently small number. Substituting this estimate to the last integral, we obtain (10). Therefore (9) holds and as a result, we have I 2 (t) = 0. It is also clear that since in consequence of Lemma 1 the integral is uniformly convergent with respect to the parameter t. Note that the last integral can be calculated as a residue, we have 1 2πi Thus, we obtain lim t→+0 1 2πi Consider a principal part of the Laurent series. Using (8), analogously to the above, we get for values n ∈ Applying Lemma 1, we obtain 1 2πi Having noticed that the integral equals to the residue of the operator function under the integral, we have 1 2πi The following reasonings are mostly the same, we get 1 2πi Applying Lemma 1, we can get easily 1 2πi Note that the principal part of the Laurent series is uniformly convergent outside a circle with a sufficiently large radius. Taking into account this fact, using the inversion, it is not hard to prove that 1 2πi Thus, we obtain the desired result.
Lemma 6. Assume that B is a compact operator, Θ(B) ⊂ L 0 (θ), θ < π/2, , the entire function ϕ is of the order less than one and of the minimal or normal type, moreover the series ∞ n=0 c n n!z n , z ∈ C is absolutely convergent, where c n are the Taylor coefficients of the function ϕ. Then Moreover, if there exists a limit ϕ(W )u(t) → g, t → +0, then g =φ(W )f.
Proof. Firstly, we should note that the conditions imposed upon the order of the function ϕ alow us to claim that the latter integral converges for a fixed value of the parameter t. Let us establish the formula 1 2πi To prove this fact, we should show that the series ∞ n=0 c n · 1 2πi Using lemma 1, we get a trivial inequality It is clear that r j r e −xt cos(θ+ε) x n dx ≤ Γ(n + 1) t n+1 cos n+1 (θ + ε) .
Thus, we get The latter relation gives us the desired result. Using formula (8), we get 1 2πi Since the operators W n and B(I −λB) −1 commute, this fact can be obtained by direct calculation, we get I 1 (t) = W n u(t). Analogously to the reasonings of Lemma 5, we get I 2 (t) = 0. Thus, we obtain the first statement of the lemma. Let us show that the operator ϕ(W ) is closeable on the set of elements In accordance with the definition (see (5.6) [3, p. 165]), we need prove that if there exist simultaneous limits u (j) (2) . Note that in accordance with Lemma 3, for each k ∈ N, we get u (j) k , t → +0. Applying the theorem which gives the connection between simultaneous limits and repeated limits, we get u (j) k → u (0) , k → ∞. Using the simple estimating, we get Therefore, there exist coincident limits Applying the theorem on the connection between simultaneous limits and repeated limits, we get u (1) = u (2) . Thus, we have proved that the operator ϕ(W ) is closeable. Applying Lemma 3, we get u(t) → f, t → +0, using condition (11), we obtain Hence f ∈ D {φ(W )} and we obtain the second statement of the lemma. The proof is complete.
Note that Corollary 1 establishes the analog of the function of a selfadjoint compact operator. The series at the right-hand side of the last formula converges to the element at the left-hand side in the (B, ϕ, 1) sense. The analog of Theorem 1 corresponding to the function with a polynomial regular part of the Laurent series considered in Lemma 5 can be obtained in a simpler way.

Introduction to evolution equations in the abstract Hilbert space
In this paragraph, we consider a Hilbert space H consists of element-functions u : R + → H, u := u(t), t ≥ 0 and we will assume that if u belongs to H then the fact holds for all values of the variable t. Notice that under such an assumption all standard topological properties as completeness, compactness e.t.c. 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 H. The derivative is understood as the following limit Let t ∈ Ω := [a, b], 0 < a < b < ∞. The following integral is understood in the Riemann sense as a limit of partial sums where (a = t 0 < t 1 < ... < t n = b) is an arbitrary splitting of the segment Ω, λ := max is an arbitrary point belonging to [t i , t i+1 ]. The sufficient condition of the last integral existence is a continuous property (see [12, p.248 Combining these operations, we can consider a fractional differential operator in the Riemann-Liouvile sense (see [24]), i.e. in the formal form, we have The auxiliary facts given above allow us to consider the following Cauchy problems as a prerequisite of a rather wide area of applications the first problem was considered in [11] under the assumption ϕ(z) = z n , n ∈ N the second one was considered in [10], where we suppose ϕ(z) = z. If we analyze the proofs of the corresponding theorems establishing the existence and uniqueness of the problems (15), then taking into account theoretical results of this paper, we will see that the corresponding generalizations can be obtained. Certainly, we should impose such conditions on the function ϕ that guarantee convergence of the obtained integral constructions. Here, we want to end our narrative and leave the idea for a further more detailed and deep study.

Conclusions
In this paper, we invented a technique to study evolution equations with the right-hand side a function of the non-selfadjoint unbounded operator. Having inspired by the Lidskii idea involving a notion of convergence of a series on the root vectors of the operator in a weaker -Abel-Lidskii sense, we continue constructing the abstract theory in the direction. The main issue of the paper is how to generalize the spectral theorem for the non-selfadjoint operator. In this way, we come to the definition of a function of the unbounded non-selfadjoint operator. The main highlights of this paper are propositions analogous to the spectral theorem, the main obstacle that appears is how to define an analogue of a spectral family or decomposition of the identical operator. We should admit that an analogous way of reformulating the main principles of the spectral theorem, considering the peculiarities of the convergence, has been found. As a prerequisite to applications, we noticed some approaches allowing us to principally broaden conditions imposed on the righthand side of the evolution equation in the abstract Hilbert space. The application part of the paper appeals to the theory of differential equations. In particular, the existence and uniqueness theorems for evolution equations with the right-hand side -a differential operator with a fractional derivative in final terms are covered by the invented abstract method. In connection with this, such operators as a Riemann-Liouville fractional differential operator, Kipriyanov operator, Riesz potential, difference operator can be considered. Moreover, we can consider the artificially constructed normal operators for which the clarification of the Lidskii results relevantly works.
Here, we should explain that we can construct a normal operator with a compact resolvent in the artificial way, having known its eigenvalues. If we analyze the proofs of the corresponding theorems establishing the existence and uniqueness of the solution of the evolution equation and take into account theoretical results of this paper, then we can see that relevant generalizations can be obtain. Apparently, we should study thoroughly conditions that should be imposed upon the function of the operator at the right-hand side of the evolution equation. This is where we want to finish in the hope that the concept will be further developed.