Abstract Evolution Equations with an Operator Function in the Second Term

Abstract

In this paper, having introduced a convergence of a series on the root vectors in the Abel-Lidskii sense, we present a valuable application to the evolution equations. The main issue of the paper is an approach allowing us to principally broaden conditions imposed upon the second term of the evolution equation in the abstract Hilbert space. In this way, we come to the definition of the function of an unbounded non-selfadjoint operator. Meanwhile, considering the main issue we involve an additional concept that is a generalization of the spectral theorem for a non-selfadjoint operator.

1. Introduction

The results connected with the application of the basic properties in the Abell-Lidskii sense [1] allow us to solve many problems [2] in the theory of evolution equations. The central idea of this paper is devoted to an approach allowing us to principally broaden conditions imposed upon the second term (in accordance with the terminology [2] “the right-hand side”) of the evolution equation in the abstract Hilbert space. In this way, we can obtain abstract results covering many applied problems to say nothing of the far-reaching generalizations. We plan to implement the idea having involved a notion of an operator function. This is why one of the paper’s challenges is to find a harmonious way of reformulating the main principles of the spectral theorem having taken into account the peculiarities of the convergence in the Abel-Lidskii sense. However, our final goal is an existence and uniqueness theorem for an abstract evolution equation with an operator function in the second term, where the derivative in the first term is supposed to be of the integer order. The peculiar result that is worth highlighting is the obtained analytic formula for the solution. We should remind that by involving a notion of the operator function, we broaden a great deal an operator class corresponding to the second term. Thus, we can state that the main issue of the paper is closely connected with the spectral theorem for the non-selfadjoint unbounded operator. Here, we should make a brief digression and consider a theoretical background that allows us to obtain such exotic results.

We should recall that the concept of the spectral theorem for a selfadjoint operator is based on the notion of a spectral family or the decomposition of the identical operator. Constructing a spectral family, we can define a selfadjoint operator using a concept of 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 a non-selfadjoint operator is not so clear but we can adopt some notions and techniques to obtain similar 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. 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 [3,4]. However, the paper [3] has a notable disadvantage regarding the conditions imposed on the numerical range of values of the operator. More precisely, a rather strong condition comparatively with the sectorial condition imposed on the numerical range of values of the operator, there considered a domain of the parabolic type containing the spectrum of the operator. A reasonable question that appears is about minimal conditions that guarantee the desired result. At the same time, the Abel-Lidskii convergence was established in the paper [5] for operators whose numerical range of values belongs to the sector with the semi-angle more than $\pi /2$. A major contribution of the paper [1] to the theory is a sufficient condition for the Abel-Lidskii basis property of the root functions system for a special type of a sectorial non-selfadjoint operator.

In the paper, [1], a peculiar technique of the entire function theory was used along with a newly introduced notion—Schatten-von Neumann class of the convergence exponent. It gave us an opportunity to invent a peculiar method of calculating the contour integrals, in its own turn the latter played a principal role in the schemes of proofs.

In order to avoid the lack of information, consider a class of non-selfadjoint operators for which the above concept can be successfully applied. We ought to note that a peculiar scientific interest appears in the case when an operator is not selfadjoint [6,7]. To be exeat, a senior term of an operator is not selfadjoint for there is a number of papers [8,9,10,11,12,13] devoted to the perturbed selfadjoint operators. It is remarkable that we can justify the methods used in the paper [6] for they have a natural mathematical origin that appears brightly while we considering abstract constructions expressed in terms of the semigroup theory [14]. We managed to study spectral properties of the operator transform applied to the infinitesimal generator and obtained the formula establishing asymptotic equivalence based on the supposedly known spectral properties of the operator real component, having applied the methods [6]. The significance is justified by the fact that the asymptotic formula for a selfadjoint operator in most cases may be obtained by virtue of the results [15]. Thus, the existence and uniqueness theorems formulated in terms of the operator order [2], subsequently generalized due to involving the notion of the operator function, can cover a significantly large operator class.

Apparently, the application part of the paper appeals to the theory of differential equations. For instance, we have an opportunity to prove the existence and uniqueness theorems for evolution equations with the second term represented by an operator function of a differential operator with a fractional derivative in final terms. Herewith, the well-known operators such as the Riemann-Liouville fractional differential operator, the Kipriyanov operator, the Riesz potential, the difference operator (more detailed [14,16,17]), the artificially constructed normal operator are involved [2]. Here, we represent a list of papers dealing with the problems which can be investigated by the obtained in this paper abstract method [18,19,20,21].

The physical significance of the problem is based on the broad field of applications. Here, we consider the most valuable example in order to show plainly the significance. Instead of using approximate methods, in the paper [22] the solution of the differential equation modeling the switching kinetics of ferroelectrics in the injection mode can be obtained in an analytical way if we impose relatively wide conditions upon the second term of the considered evolution equation.

2. Preliminaries

Throughout the paper, if the contrary is not stated, we consider linear densely defined operators acting on a separable complex Hilbert space $\mathfrak{H}$.

Below, we present a list of notations.

• C—a real positive constant, we assume that a value of C can be different within a formula.
• $\mathcal{B}\left(\mathfrak{H}\right)$—the set of linear bounded operators on $\mathfrak{H}.$
• $\mathrm{D}\left(T\right)$—the domain of definition of the operator $T.$
• $\mathrm{R}\left(T\right)$—the range of the operator $T.$
• $\mathrm{N}\left(T\right)$—the kernel of the operator $T.$
• $\mathrm{P}\left(T\right)$—the resolvent set of the operator $T.$
• ${\lambda }_i\left(T\right),i\in \mathbb{N}$—the eigenvalues of the operator $T.$
• ${\lambda }_i\left(T\right),i\in \mathbb{N}$—the eigenvalues of the operator $T.$
• $s_i\left(T\right),i=1,2,\dots ,r\left(S\right)$—the singular numbers (s-numbers) of the compact operator $T,$ where $r\left(S\right):=\dim \mathrm{R}\left(S\right),S:={\left(T^{\ast }T\right)}^{1/2}.$
• $n\left(r\right)$—the function equals to the number of the elements of the sequence ${\left\{a_n\right\}}_1^{\infty },|a_n|\uparrow \infty$ within the circle $\left|z\right|<r.$
• $n_T\left(r\right)$—the counting function a function $n\left(r\right)$ corresponding to the sequence ${\left\{s_i^{-1}\left(T\right)\right\}}_1^{\infty },$ where T is a compact operator.
• ${\mathfrak{S}}_p\left(\mathfrak{H}\right),0<p<\infty$—the Schatten-von Neumann class.
• ${\mathfrak{S}}_{\infty }\left(\mathfrak{H}\right)$—the set of compact operators.
• ${\tilde{\mathfrak{S}}}_{\rho }\left(\mathfrak{H}\right)$—the Schatten-von Neumann class of the convergence exponent, defined as follows ${\tilde{\mathfrak{S}}}_{\rho }\left(\mathfrak{H}\right):=\{T\in {\mathfrak{S}}_{\rho +\epsilon },T\overline{\in }{\mathfrak{S}}_{\rho -\epsilon },\forall \epsilon >0,\rho <\infty \}.$
• $\mathsf{\Theta }\left(T\right)$—the numerical range of the operator $T,$ defined as follows $\mathsf{\Theta }\left(T\right):=\{z\in \mathbb{C}:z={(Th,h)}_{\mathfrak{H}},h\in \mathrm{D}\left(T\right),\parallel h{\parallel }_{\mathfrak{H}}=1\}.$
• $\mathrm{int}\mathfrak{M}$— the interior of the set $\mathfrak{M}.$
• $\mathrm{Fr}\mathfrak{M}$—the set of boundary points of the set $\mathfrak{M}.$
• ${\mathfrak{L}}_{\iota }\left(\theta \right)$—a closed sector with the vertex $\iota$ and the semi-angle of the sector $\theta ,$ defined as follows ${\mathfrak{L}}_{\iota }\left(\theta \right):=\{z:|arg(z-\iota )|\le \theta <\pi /2\}.$ If we want to stress the correspondence between $\iota$ and $\theta ,$ then we will write ${\theta }_{\iota }.$
• $M_f\left(r\right)$— the maximum absolute value of the function f on the boundary of the circle with the radios r and the center at point zero.
• $m_f\left(r\right)$— the minimum absolute value of the function f on the boundary of the circle with the radios r and the center at point zero.

By the convergence exponent $\rho$ of the sequence ${\left\{a_n\right\}}_1^{\infty }\subset \mathbb{C},a_n\ne 0,a_n\to \infty$ we mean the greatest lower bound for such numbers $\lambda$ that the following series converges

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

More detailed information can be found in [23]. Throughout the paper, unless otherwise stated, we use notations of the papers [4,24].

• Convergence in the Abel-Lidsky sense

In this paragraph, we have made an attempt to represent results [5] in order to make them more applicable to the aims of this paper. Using the fact that the spectrum of an arbitrary compact operator T consists of the so-called normal eigenvalues (in accordance with the Hilbert theorem ([4], p. 32), [25]), consider a direct sum of subspaces

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

(1)

where the summands are invariant subspaces of the operator T (more detailed information see [1]). Let $n_q$ be a dimension of ${\mathfrak{N}}_q$ and let $T_q$ be the operator induced in ${\mathfrak{N}}_q.$ We can choose a Jordan basis in ${\mathfrak{N}}_q$ that consists of Jordan chains of eigenvectors and root vectors of the operator $T_q.$ Each chain $e_{q_{\xi }},e_{q_{\xi }+1},\dots ,e_{q_{\xi }+k},k\in {\mathbb{N}}_0,$ where $e_{q_{\xi }},\xi =1,2,\dots ,m$ are the eigenvectors corresponding to the eigenvalue ${\mu }_q$ and other terms are root vectors, can be transformed by the operator T in accordance with the following formulas

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

(2)

Considering the sequence ${\left\{{\mu }_q\right\}}_1^{\infty }$ of the eigenvalues of the operator T and choosing a Jordan basis in each corresponding space ${\mathfrak{N}}_q,$ we can arrange a system of vectors ${\left\{e_i\right\}}_1^{\infty }$ which we will call a system of the root vectors or following Lidskii a system of the major vectors of the operator $T.$ Assume that $e_1,e_2,\dots ,e_{n_q}$ is the Jordan basis in the subspace ${\mathfrak{N}}_q.$ We can prove easily (see [5], p. 14) that there exists a corresponding biorthogonal basis $g_1,g_2,\dots ,g_{n_q}$ in the subspace ${\mathfrak{M}}_q^{\perp }.$

Using the reasonings [1], we conclude that ${\left\{g_i\right\}}_1^{n_q}$ consists of the Jordan chains of the operator $T^{\ast }$ which correspond to the Jordan chains (2) due to the following formula $T^{\ast }{g}_{q_{\xi }+k}={\overline{\mu }}_q{g}_{q_{\xi }+k},T^{\ast }{g}_{q_{\xi }+k-1}={\overline{\mu }}_q{g}_{q_{\xi }+k-1}+g_{q_{\xi }+k},\dots ,T^{\ast }{g}_{q_{\xi }}={\overline{\mu }}_q{g}_{q_{\xi }}+g_{q_{\xi }+1}.$ It is not hard to prove that the set ${\left\{g_{\nu }\right\}}_1^{n_j},j\ne i$ is orthogonal to the set ${\left\{e_{\nu }\right\}}_1^{n_i}$ (see [1]). Gathering the sets ${\left\{g_{\nu }\right\}}_1^{n_j},j=1,2,\dots ,$ we can obviously create a biorthogonal system ${\left\{g_n\right\}}_1^{\infty }$ with respect to the system of the major vectors of the operator $T.$ It is rather reasonable to call it as a system of the major vectors of the operator $T^{\ast }.$

Let us come to the previously made agreement that the vectors in each Jordan chain are arranged in the same order as in (2) i.e., in the first place there stands an eigenvector. It is clear that under such an assumption we have $c_{q_{\xi }+i}=(f,g_{q_{\xi }+k-i})/(e_{q_{\xi }+i},g_{q_{\xi }+k-i}),$ $0\le i\le k\left(q_{\xi }\right),$ where $k\left(q_{\xi }\right)+1$ is a number of elements in the $q_{\xi }$-th Jourdan chain. In particular, if the vector $e_{q_{\xi }}$ is included in the major system solo, there does not exist a root vector corresponding to the same eigenvalue, then $c_{q_{\xi }}=(f,g_{q_{\xi }})/(e_{q_{\xi }},g_{q_{\xi }}).$ Define a formal set of functions depending on a real parameter t

$$H_m(\phi ,z,t):=\frac{e^{\phi \left(z\right)t}}{m!}·\frac{d^m}{d{z}^m}\left\{e^{-\phi \left(z\right)t}\right\},m=0,1,2,\dots .$$

Here we should note that if $\phi :=z,z\in \mathbb{C},$ then we have a set of polynomials, what is in the origin of the concept, see [5]. Consider a series

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

(3)

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

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

(4)

here ${\lambda }_q=1/{\mu }_q$ is a characteristic number corresponding to $e_{q_{\xi }}.$ It is clear that in any case, we have a limit $c_n\left(t\right)\to {\tilde{c}}_n,t\to +0,$ where a value ${\tilde{c}}_n$ can be calculated directly due to the Formula (4). For instance in the case $\phi =z,z\in \mathbb{C},$ we have ${\tilde{c}}_n=c_n.$ Generalizing the definition given in ([26], p. 71), we will say that series (3) converges to the element f in the sense $(A,\lambda ,\phi ),$ if there exists a sequence of the natural numbers ${\left\{N_j\right\}}_1^{\infty }$ such that

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

We need the following lemmas [5], in the adopted form also see [1]. In spite of the fact that the scheme of the Lemma 3 proof is the same, we present it in expanded form for the reader’s convenience. Suppose a compact operator $T:\mathfrak{H}\to \mathfrak{H}$ such that $\mathsf{\Theta }\left(T\right)\subset {\mathfrak{L}}_0\left(\theta \right),\theta <\pi .$ Then, we put the following contour in correspondence to the operator

$$\vartheta \left(T\right):=\left\{\lambda :\left|\lambda \right|=r>0,\left|\arg \lambda \right|\le \theta +\varsigma \right\}\cup \left\{\lambda :\left|\lambda \right|>r,\left|\arg \lambda \right|=\theta +\varsigma \right\},$$

where $\varsigma >0$ is an arbitrary small number, the number r is chosen so that the operator ${(I-\lambda T)}^{-1}$ is regular within the corresponding closed circle.

Lemma 1.

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

$${\parallel (I-\lambda T)}^{-1}\parallel \le \frac{1}{sin{\psi }_0},\lambda \in \zeta ,$$

where ${\psi }_0=min\left\{\right|\arg \zeta -\theta |,|\arg \zeta +\theta \left|\right\}.$

Lemma 2.

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

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

where

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

Lemma 3.

Assume that T is a compact operator, φ is an analytical function inside $\vartheta \left(T\right),$ then in the pole ${\lambda }_q$ of the operator ${(I-\lambda T)}^{-1},$ the residue of the vector function $e^{-\phi \left(\lambda \right)t}T{(I-\lambda T)}^{-1}f,$$(f\in \mathfrak{H}),$ equals to

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

where $m\left(q\right)$ is a geometrical multiplicity of the q-th eigenvalue, $k\left(q_{\xi }\right)+1$ is a number of elements in the $q_{\xi }$-th Jourdan chain,

$$c_{q_{\xi }+j}\left(t\right):=e^{-\phi \left({\lambda }_q\right)t}\sum _{m=0}^{k\left(q_{\xi }\right)-j}{c}_{q_{\xi }+j+m}{H}_m(\phi ,{\lambda }_q,t).$$

Proof. 

Consider an integral

$$\mathfrak{I}=\frac{1}{2\pi i}\underset{{\vartheta }_q}{\oint }{e}^{-\phi \left(\lambda \right)t}T{(I-\lambda T)}^{-1}fd\lambda ,f\in \mathrm{R}\left(T\right),$$

where the interior of the contour ${\vartheta }_q$ does not contain any poles of the operator ${(I-\lambda T)}^{-1},$ except of ${\lambda }_q.$ Assume that ${\mathfrak{N}}_q$ is a root space corresponding to ${\lambda }_q$ and consider a Jordan basis {$e_{q_{\xi }+i}\},i=0,1,\dots ,k\left(q_{\xi }\right),\xi =1,2,\dots ,m\left(q\right)$ in ${\mathfrak{N}}_q.$ Using decomposition of the Hilbert space in the direct sum (1), we can represent an element

$$f=f_1+f_2,$$

where $f_1\in {\mathfrak{N}}_q,f_2\in {\mathfrak{M}}_q.$ Note that the operator function $e^{-\phi \left(\lambda \right)t}T{(I-\lambda T)}^{-1}{f}_2$ is regular in the interior of the contour ${\vartheta }_q,$ it follows from the fact that ${\lambda }_q$ ia a normal eigenvalue. Hence, we have

$$\mathfrak{I}=\frac{1}{2\pi i}\underset{{\vartheta }_q}{\oint }{e}^{-\phi \left(\lambda \right)t}T{(I-\lambda T)}^{-1}{f}_{1}d\lambda .$$

Using the formula

$$T{(I-\lambda T)}^{-1}=\left\{{(I-\lambda T)}^{-1}-I\right\}\frac{1}{\lambda }=\left\{{\left(\frac{1}{\lambda }I-T\right)}^{-1}-\lambda I\right\}\frac{1}{{\lambda }^2},$$

we obtain

$$\mathfrak{I}=-\frac{1}{2\pi i}\underset{{\tilde{\vartheta }}_q}{\oint }{e}^{-\phi \left({\zeta }^{-1}\right)t}T{(\zeta I-T)}^{-1}{f}_{1}d\zeta ,\zeta =1/\lambda .$$

Now, let us decompose the element $f_1$ on the corresponding Jordan basis, we have

$$f_1=\sum _{\xi =1}^{m\left(q\right)}\sum _{i=0}^{k\left(q_{\xi }\right)}{e}_{q_{\xi }+i}{c}_{q_{\xi }+i}.$$

(5)

In accordance with the relation (2), we get

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

Using this formula, we can prove the following relation

$${(\zeta I-T)}^{-1}{e}_{q_{\xi }+i}=\sum _{j=0}^i\frac{e_{q_{\xi }+j}}{{(\zeta -{\mu }_q)}^{i-j+1}}.$$

(6)

Note that the case $i=0$ is trivial. Consider a case, when $i>0,$ we have

$$\frac{(\zeta I-T)e_{q_{\xi }+j}}{{(\zeta -{\mu }_q)}^{i-j+1}}=\frac{\zeta e_{q_{\xi }+j}-T{e}_{q_{\xi }+j}}{{(\zeta -{\mu }_q)}^{i-j+1}}=\frac{e_{q_{\xi }+j}}{{(\zeta -{\mu }_q)}^{i-j}}-\frac{e_{q_{\xi }+j-1}}{{(\zeta -{\mu }_q)}^{i-j+1}},j>0,$$

$$\frac{(\zeta I-T)e_{q_{\xi }}}{{(\zeta -{\mu }_q)}^{i+1}}=\frac{e_{q_{\xi }}}{{(\zeta -{\mu }_q)}^i}.$$

Using these formulas, we obtain

$$\sum _{j=0}^i\frac{(\zeta I-T)e_{q_{\xi }+j}}{{(\zeta -{\mu }_q)}^{i-j+1}}=\frac{e_{q_{\xi }}}{{(\zeta -{\mu }_q)}^i}+\frac{e_{q_{\xi }+1}}{{(\zeta -{\mu }_q)}^{i-1}}-\frac{e_{q_{\xi }}}{{(\zeta -{\mu }_q)}^i}+\dots$$

$$+\frac{e_{q_{\xi }+i}}{{(\zeta -{\mu }_q)}^{i-i}}-\frac{e_{q_{\xi }+i-1}}{{(\zeta -{\mu }_q)}^{i-i+1}}=\frac{e_{q_{\xi }+i}}{{(\zeta -{\mu }_q)}^{i-i}},$$

what gives us the desired result. Now, substituting (5) and (6), we get

$$\mathfrak{I}=-\frac{1}{2\pi i}\sum _{\xi =1}^{m\left(q\right)}\sum _{i=0}^{k\left(q_{\xi }\right)}{c}_{q_{\xi }+i}\sum _{j=0}^i{e}_{q_{\xi }+j}\underset{{\tilde{\vartheta }}_q}{\oint }\frac{e^{-\phi \left({\zeta }^{-1}\right)t}}{{(\zeta -{\mu }_q)}^{i-j+1}}d\zeta .$$

Note that the function $\phi \left({\zeta }^{-1}\right)$ is analytic inside the interior of ${\tilde{\vartheta }}_q,$ hence

$$\frac{1}{2\pi i}\underset{{\tilde{\vartheta }}_q}{\oint }\frac{e^{-\phi \left({\zeta }^{-1}\right)t}}{{(\zeta -{\mu }_q)}^{i-j+1}}d\zeta =\frac{1}{(i-j)!}\underset{\zeta \to {\mu }_q}{lim}\frac{d^{i-j}}{d{\zeta }^{i-j}}\left\{e^{-\phi \left({\zeta }^{-1}\right)t}\right\}=:e^{-\phi \left({\lambda }_q\right)t}{H}_{i-j}(\phi ,{\lambda }_q,t).$$

Changing the indexes, we have

$$\mathfrak{I}=-\sum _{\xi =1}^{m\left(q\right)}\sum _{i=0}^{k\left(q_{\xi }\right)}{c}_{q_{\xi }+i}{e}^{-\phi \left({\lambda }_q\right)t}\sum _{j=0}^i{e}_{q_{\xi }+j}{H}_{i-j}(\phi ,{\lambda }_q,t)=-\sum _{\xi =1}^{m\left(q\right)}\sum _{j=0}^{k\left(q_{\xi }\right)}{e}_{q_{\xi }+j}{e}^{-\phi \left({\lambda }_q\right)t}\sum _{m=0}^{k\left(q_{\xi }\right)-j}{c}_{q_{\xi }+j+m}{H}_m(\phi ,{\lambda }_q,t)$$

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

where

$$c_{q_{\xi }+j}\left(t\right):=e^{-\phi \left({\lambda }_q\right)t}\sum _{m=0}^{k\left(q_{\xi }\right)-j}{c}_{q_{\xi }+j+m}{H}_m(\phi ,{\lambda }_q,t).$$

The proof is complete. □

Note that using the reasonings of the last lemma, it is not hard to prove that

$$c_{q_{\xi }+j}\left(t\right)⟶\sum _{m=0}^{k\left(q_{\xi }\right)-j}{c}_{q_{\xi }+j+m}{H}_m(\phi ,{\lambda }_q,0),t\to +0.$$

3. Main Results

In this section, we have a challenge in how to generalize results [1] in the way to make an efficient tool for studying abstract evolution equations with the operator function in the second term. The operator function is supposed to be defined on the set of unbounded non-selfadjoint operators. First of all, we consider statements with the necessary refinement caused by the involved functions, here we should note that a particular case corresponding to a power function $\phi$ was considered by Lidskii [5]. Secondly, we find conditions that guarantee convergence of the involved integral construction and formulate lemmas giving us a tool for further study. We consider an analog of the spectral family having involved the operators similar to Riesz projectors (see [4], p. 20) and using a notion of the convergence in the Abel-Lidskii sense. As a main result, we prove an existence and uniqueness theorem for the evolution equation with the operator function in the second term. Finally, we discuss an approach that we can implement to apply the abstract theoretical results to concrete evolution equations.

Lemma 4.

Suppose the operator B satisfies conditions of Lemma 1, the entire function φ of the order less than a half maps the inside of the contour $\vartheta \left(B\right)$ into the sector ${\mathfrak{L}}_0\left(\varpi \right),\varpi <\pi /2$ for a sufficiently large value $\left|z\right|,z\in \mathrm{int}\vartheta \left(B\right).$ Then the following relation holds

$$\underset{t\to +0}{lim}\frac{1}{2\pi i}\underset{\vartheta \left(B\right)}{\int }{e}^{-\phi \left(\lambda \right)t}B{(I-\lambda B)}^{-1}fd\lambda =f,f\in \mathrm{R}\left(B\right).$$

(7)

Proof. 

Using the formula

$$B^2{(I-\lambda B)}^{-1}=\frac{1}{{\lambda }^2}\left\{{\left(I-\lambda B\right)}^{-1}-(I+\lambda B)\right\},$$

we obtain

$$\frac{1}{2\pi i}\underset{\vartheta \left(B\right)}{\int }{e}^{-\phi \left(\lambda \right)t}B{(I-\lambda B)}^{-1}fd\lambda =\frac{1}{2\pi i}\underset{\vartheta \left(B\right)}{\int }{e}^{-\phi \left(\lambda \right)t}{\lambda }^{-2}{\left(I-\lambda B\right)}^{-1}Wfd\lambda$$

$$-\frac{1}{2\pi i}\underset{\vartheta \left(B\right)}{\int }{e}^{-\phi \left(\lambda \right)t}{\lambda }^{-2}(I+\lambda B)Wfd\lambda =I_1\left(t\right)+I_2\left(t\right).$$

Consider $I_1\left(t\right).$ Since this improper integral is uniformly convergent regarding $t,$ this fact can be established easily if we apply Lemma 1, then using the theorem on the connection with the simultaneous limit and the repeated limit, we get

$$\underset{t\to +0}{lim}{I}_1\left(t\right)=\frac{1}{2\pi i}\underset{\vartheta \left(B\right)}{\int }{\lambda }^{-2}{\left(I-\lambda B\right)}^{-1}Wfd\lambda .$$

define a contour ${\vartheta }_R\left(B\right):=\mathrm{Fr}\left\{\{\lambda :|\lambda |<R\}\backslash \mathrm{int}\vartheta \left(B\right)\}\right\}$ and let us prove that

$$\frac{1}{2\pi i}\underset{{\vartheta }_R\left(B\right)}{\oint }{\lambda }^{-2}{\left(I-\lambda B\right)}^{-1}Wfd\lambda \to \frac{1}{2\pi i}\underset{\vartheta \left(B\right)}{\int }{\lambda }^{-2}{\left(I-\lambda B\right)}^{-1}Wfd\lambda ,R\to \infty .$$

(8)

Consider a decomposition of the contour ${\vartheta }_R\left(B\right)$ on terms ${\tilde{\vartheta }}_R\left(B\right):=\{\lambda :|\lambda |=R,\theta +\varsigma \le \arg \lambda \le 2\pi -\theta -\varsigma \},{\widehat{\vartheta }}_R:=\{\lambda :|\lambda |=r,|\arg \lambda |\le \theta +\varsigma \}\cup \{\lambda :r<|\lambda |<R,\arg \lambda =\theta +\varsigma \}\cup \{\lambda :r<|\lambda |<R,\arg \lambda =-\theta -\varsigma \}.$ It is clear that

$$\frac{1}{2\pi i}\underset{{\vartheta }_R\left(B\right)}{\oint }{\lambda }^{-2}{\left(I-\lambda B\right)}^{-1}Wfd\lambda =\frac{1}{2\pi i}\underset{{\tilde{\vartheta }}_R\left(B\right)}{\int }{\lambda }^{-2}{\left(I-\lambda B\right)}^{-1}Wfd\lambda$$

$$+\frac{1}{2\pi i}\underset{{\widehat{\vartheta }}_R}{\int }{\lambda }^{-2}{\left(I-\lambda B\right)}^{-1}Wfd\lambda .$$

Let us show that the first summand tends to zero when $R\to \infty ,$ we have

$${∥\underset{{\tilde{\vartheta }}_R\left(B\right)}{\int }{\lambda }^{-2}{\left(I-\lambda B\right)}^{-1}Wfd\lambda ∥}_{\mathfrak{H}}\le R^{-2}\underset{\theta +\varsigma }{\overset{2\pi -\theta -\varsigma }{\int }}{∥{\left(I{\lambda }^{-1}-B\right)}^{-1}Wf∥}_{\mathfrak{H}}d\arg \lambda .$$

Applying Corollary 3.3, Theorem 3.2 ([24], p. 268), we have

$${∥{\left(I{\lambda }^{-1}-B\right)}^{-1}∥}_{\mathfrak{H}}\le R/sin\varsigma ,\lambda \in {\tilde{\vartheta }}_R\left(B\right).$$

Substituting this estimate to the last integral, we obtain the desired result. Thus, taking into account the fact

$$\frac{1}{2\pi i}\underset{{\widehat{\vartheta }}_R}{\int }{\lambda }^{-2}{\left(I-\lambda B\right)}^{-1}Wfd\lambda \to \frac{1}{2\pi i}\underset{\vartheta \left(B\right)}{\int }{\lambda }^{-2}{\left(I-\lambda B\right)}^{-1}Wfd\lambda ,R\to \infty ,$$

we obtain (8). Having noticed that the following integral can be calculated as a residue at the point zero, i.e.,

$$\frac{1}{2\pi i}\underset{{\vartheta }_R\left(B\right)}{\oint }{\lambda }^{-2}{\left(I-\lambda B\right)}^{-1}Wfd\lambda =\underset{\lambda \to 0}{lim}\frac{d{(I-\lambda B)}^{-1}}{d\lambda }Wf=f,$$

we get

$$\frac{1}{2\pi i}\underset{\vartheta \left(B\right)}{\int }{\lambda }^{-2}{\left(I-\lambda B\right)}^{-1}Wfd\lambda =f.$$

Hence $I_1\left(t\right)\to f,t\to +0.$ Let us show that $I_2\left(t\right)=0.$ For this purpose, let us consider a contour ${\vartheta }_R\left(B\right)={\tilde{\vartheta }}_R\cup {\widehat{\vartheta }}_R,$ where ${\tilde{\vartheta }}_R:=\{\lambda :|\lambda |=R,|\arg \lambda |\le \theta +\varsigma \}$ and ${\widehat{\vartheta }}_R$ is previously defined. It is clear that

$$\frac{1}{2\pi i}\underset{{\vartheta }_R\left(B\right)}{\oint }{\lambda }^{-2}{e}^{-\phi \left(\lambda \right)t}\left(I+\lambda B\right)Wfd\lambda =\frac{1}{2\pi i}\underset{{\tilde{\vartheta }}_R}{\int }{\lambda }^{-2}{e}^{-\phi \left(\lambda \right)t}\left(I+\lambda B\right)Wfd\lambda$$

$$+\frac{1}{2\pi i}\underset{{\widehat{\vartheta }}_R}{\int }{\lambda }^{-2}{e}^{-\phi \left(\lambda \right)t}\left(I+\lambda B\right)Wfd\lambda .$$

Considering the second term having taken into account the definition of the improper integral, we conclude that if we show that there exists such a sequence ${\left\{R_n\right\}}_1^{\infty },R_n\uparrow \infty$ that

$$\frac{1}{2\pi i}\underset{{\tilde{\vartheta }}_{R_n}}{\int }{\lambda }^{-2}{e}^{-\phi \left(\lambda \right)t}\left(I+\lambda B\right)Wfd\lambda \to 0,n\to \infty ,$$

(9)

then we obtain

$$\frac{1}{2\pi i}\underset{{\vartheta }_{R_n}\left(B\right)}{\oint }{\lambda }^{-2}{e}^{-\phi \left(\lambda \right)t}\left(I+\lambda B\right)Wfd\lambda \to \frac{1}{2\pi i}\underset{\vartheta \left(B\right)}{\int }{\lambda }^{-2}{e}^{-\phi \left(\lambda \right)t}\left(I+\lambda B\right)Wfd\lambda ,R\to \infty .$$

(10)

Using the lemma conditions, we can accomplish the following estimation

$$|e^{-\phi \left(\lambda \right)t}|=e^{-\mathrm{Re}\phi \left(\lambda \right)t}\le e^{-C\left|\phi \right(\lambda \left)\right|t},\lambda \in {\tilde{\vartheta }}_R,$$

(11)

where R is sufficiently large. Using the condition imposed upon the order of the entire function and applying the Wieman theorem (Theorem 30 §18 Chapter I [23]), we can claim that there exists such a sequence ${\left\{R_n\right\}}_1^{\infty },R_n\uparrow \infty$ that

$$\forall \epsilon >0,\exists N\left(\epsilon \right):e^{-C\left|\phi \right(\lambda \left)\right|t}\le e^{-C{m}_{\phi }\left(R_n\right)t}\le e^{-Ct{\left[M_{\phi }\left(R_n\right)\right]}^{cos\pi \varphi -\epsilon }},\lambda \in {\tilde{\vartheta }}_{R_n},n>N\left(\epsilon \right),$$

where $\varphi$ is the order of the entire function $\phi .$ Using this estimate, we get

$${∥\underset{{\tilde{\vartheta }}_{R_n}}{\int }{\lambda }^{-2}{e}^{-\phi \left(\lambda \right)t}\left(I+\lambda B\right)Wfd\lambda ∥}_{\mathfrak{H}}\le C{e}^{-Ct{\left[M_{\phi }\left(R_n\right)\right]}^{cos\pi \varphi -\epsilon }}{\parallel Wf\parallel }_{\mathfrak{H}}\underset{-\theta -\varsigma }{\overset{\theta +\varsigma }{\int }}d\xi .$$

It is clear that if the order $\varphi$ is less than half, then we obtain (9) and as a consequence (10). Since the operator function under the integral is analytic, then

$$\underset{{\vartheta }_{R_n}\left(B\right)}{\oint }{\lambda }^{-2}{e}^{-\phi \left(\lambda \right)t}\left(I+\lambda B\right)Wfd\lambda =0,n\in \mathbb{N}.$$

Combining this relation with (10), we obtain the fact $I_2\left(t\right)=0.$ The proof is complete. □

Remark 1.

Note that the statement of the lemma is not true if the order equals zero, in this case, we cannot apply the Wieman theorem (for more detail, see the proof of the Theorem 30 §18 Chapter I [23]). At the same time, the proof can be easily transformed for the case corresponding to a polynomial function. Here, we should note that the reasonings are the same, we have to impose conditions upon the polynomial to satisfy the lemma conditions and establish an estimate analogous to (11). Now assume that $\phi \left(z\right)=c_0+c_{1}z+\cdots +c_n{z}^n,z\in \mathbb{C},$ by easy calculations we see that the condition

$$\underset{k=0,1,\dots ,n}{max}\left(\right|\arg c_k|+k\theta )<\pi /2,$$

gives us $\left|\arg \phi \right(z\left)\right|<\pi /2,z\in \mathrm{int}\vartheta \left(B\right).$ Thus, we have the fulfillment of the estimate (11). It can be established easily that $m_{\phi }\left(\right|z\left|\right)\to \infty ,\left|z\right|\to \infty .$ Combining this fact with (11) and preserving the scheme of the reasonings presented in Lemma 4, we obtain (7).

Bellow, we consider an invertible operator B and use a notation $W:=B^{-1}.$ This agreement is justified by the significance of the operator with a compact resolvent, the detailed information on which spectral properties can be found in the papers cited in the introduction section. Consider a function $\phi$ that can be represented by a Laurent series about point zero. Denote by

$$\phi \left(W\right):=\sum _{n=-\infty }^{\infty }{c}_n{W}^n$$

(12)

a formal construction called by a function of the operator, where $c_n$ are coefficients corresponding to the function $\phi .$ The lemma given below is devoted to the study of the conditions under which being imposed the series of operators (12) converges on some elements of the Hilbert space $\mathfrak{H},$ thus the operator $\phi \left(W\right)$ is defined.

Lemma 5.

Suppose B is a compact operator, $\mathsf{\Theta }\left(B\right)\subset {\mathfrak{L}}_0\left(\theta \right),\theta <\pi /2,$

$$\phi \left(z\right)=\sum _{n=-\infty }^s{c}_n{z}^n,z\in \mathbb{C},s\in \mathbb{N},\underset{n=0,1,\dots ,s}{max}\left(\right|\arg c_n|+n\theta )<\pi /2,$$

(13)

then

$$\frac{1}{2\pi i}\underset{\vartheta \left(B\right)}{\int }\phi \left(\lambda \right)e^{-\phi \left(\lambda \right)t}B{(I-\lambda B)}^{-1}fd\lambda =\phi \left(W\right)u\left(t\right);\underset{t\to +0}{lim}\phi \left(W\right)u\left(t\right)=\phi \left(W\right)f,$$

(14)

where

$$u\left(t\right):=\frac{1}{2\pi i}\underset{\vartheta \left(B\right)}{\int }{e}^{-\phi \left(\lambda \right)t}B{(I-\lambda B)}^{-1}fd\lambda ,f\in \mathrm{D}\left(W^s\right).$$

Proof. 

Consider a decomposition of the Laurent series on two terms

$${\phi }_1\left(z\right)=\sum _{n=0}^s{c}_n{z}^n;{\phi }_2\left(z\right)=\sum _{n=1}^{\infty }{c}_{-n}{z}^{-n}.$$

Consider an obvious relation

$${\lambda }^k{B}^k{(E-\lambda B)}^{-1}={(E-\lambda B)}^{-1}-(E+\lambda B+\cdots +{\lambda }^{k-1}{B}^{k-1}),k\in \mathbb{N}.$$

(15)

It gives us the following representation

$$\frac{1}{2\pi i}\underset{\vartheta \left(B\right)}{\int }{\lambda }^n{e}^{-\phi \left(\lambda \right)t}B{(I-\lambda B)}^{-1}fd\lambda =I_{1n}\left(t\right)+I_{2n}\left(t\right),n\in {\mathbb{Z}}^{-}\cup \{0,1,\dots ,s\},$$

(16)

where

$$I_{1n}:=\frac{1}{2\pi i}\underset{\vartheta \left(B\right)}{\int }{e}^{-\phi \left(\lambda \right)t}{(I-\lambda B)}^{-1}{W}^{n-1}fd\lambda ,I_{2n}\left(t\right):=0,n=0,$$

$$I_{2n}\left(t\right):=\left\{\begin{array}{c}\hfill -\sum _{k=0}^{n-1}{\beta }_k\left(t\right)B^{k-n+1}f,n>0,\\ \hfill \sum _{k=-1}^n{\beta }_k\left(t\right)B^{k-n+1}f,n<0\end{array}\right.,{\beta }_k\left(t\right):=\frac{1}{2\pi i}\underset{\vartheta \left(B\right)}{\int }{e}^{-\phi \left(\lambda \right)t}{\lambda }^{k}d\lambda .$$

Let us show that ${\beta }_k\left(t\right)=0,$ define a contour ${\vartheta }_R\left(B\right):=\mathrm{Fr}\left\{\mathrm{int}\vartheta \left(B\right)\cap \{\lambda :r<|\lambda |<R\}\right\}$ and let us prove that

$$I_{Rk}\left(t\right):=\frac{1}{2\pi i}\underset{{\vartheta }_R\left(B\right)}{\oint }{e}^{-\phi \left(\lambda \right)t}{\lambda }^{k}d\lambda \to {\beta }_k\left(t\right),R\to \infty .$$

(17)

Consider a decomposition of the contour ${\vartheta }_R\left(B\right)$ on terms ${\tilde{\vartheta }}_R:=\{\lambda :|\lambda |=R,|\arg \lambda |\le \theta +\varsigma \}$ and ${\widehat{\vartheta }}_R:=\{\lambda :|\lambda |=r,|\arg \lambda |\le \theta +\varsigma \}\cup \{\lambda :r<|\lambda |<R,\arg \lambda =\theta +\varsigma \}\cup \{\lambda :r<|\lambda |<R,\arg \lambda =-\theta -\varsigma \}.$ We have

$$\frac{1}{2\pi i}\underset{{\vartheta }_R\left(B\right)}{\oint }{e}^{-\phi \left(\lambda \right)t}{\lambda }^{k}d\lambda =\frac{1}{2\pi i}\underset{{\tilde{\vartheta }}_R}{\int }{e}^{-\phi \left(\lambda \right)t}{\lambda }^{k}d\lambda +\frac{1}{2\pi i}\underset{{\widehat{\vartheta }}_R}{\int }{e}^{-\phi \left(\lambda \right)t}{\lambda }^{k}d\lambda .$$

Having noticed that $I_{Rk}\left(t\right)=0,$ since the operator function under the integral is analytic inside the contour, we come to the conclusion that to obtain the desired result, we should show

$$\frac{1}{2\pi i}\underset{{\tilde{\vartheta }}_R}{\int }{e}^{-\phi \left(\lambda \right)t}{\lambda }^{k}d\lambda \to 0,R\to \infty .$$

(18)

We have

$$\left|\underset{{\tilde{\vartheta }}_R}{\int }{e}^{-\phi \left(\lambda \right)t}{\lambda }^{k}d\lambda \right|\le R^k\underset{{\tilde{\vartheta }}_R}{\int }|e^{-\phi \left(\lambda \right)t}\left|\right|d\lambda |\le R^{k+1}\underset{-\theta -\varsigma }{\overset{\theta +\varsigma }{\int }}{e}^{-t\mathrm{Re}\phi \left(\lambda \right)}d\arg \lambda .$$

Consider a value $\mathrm{Re}\phi \left(\lambda \right),\lambda \in {\tilde{\vartheta }}_R$ for a sufficiently large value $R.$ Using the property of the principal part of the Laurent series in is not hard to prove that $\forall \epsilon >0,\exists N\left(\epsilon \right):|{\phi }_2\left(\lambda \right)|<\epsilon ,R>N\left(\epsilon \right).$ It follows easily from the condition (13) that $\mathrm{Re}{\phi }_1\left(\lambda \right)\ge C\left|{\phi }_1\left(\lambda \right)\right|,\lambda \in {\tilde{\vartheta }}_R.$ It is clear that $|{\phi }_1\left(\lambda \right)|\sim |c_s|R^s,R\to \infty .$ Thus, we have

$$e^{-t\mathrm{Re}\phi \left(\lambda \right)}\le e^{-Ct\left|\phi \right(\lambda \left)\right|}\le e^{-{C\left|\lambda \right|}^{s}t},\lambda \in {\tilde{\vartheta }}_R.$$

(19)

Applying this estimate, we obtain

$$\underset{-\theta -\varsigma }{\overset{\theta +\varsigma }{\int }}{e}^{-t\mathrm{Re}\phi \left(\lambda \right)}d\arg \lambda \le \underset{-\theta -\varsigma }{\overset{\theta +\varsigma }{\int }}{e}^{-Ct\left|\phi \right(\lambda \left)\right|}d\arg \lambda \le e^{-Ct{R}^s}\underset{-\theta -\varsigma }{\overset{\theta +\varsigma }{\int }}d\arg \lambda .$$

The latter estimate gives us (18) from what follows (17). Therefore ${\beta }_k\left(t\right)=0$ and we obtain the fact $I_{2n}\left(t\right)=0.$ Combining the fact of the operator W closedness (see [24], p. 165) with the definition of the integral in the Riemann sense, we get easily

$$W^{n}u\left(t\right)=\frac{1}{2\pi i}\underset{\vartheta \left(B\right)}{\int }{e}^{-\phi \left(\lambda \right)t}B{(I-\lambda B)}^{-1}{W}^{n}fd\lambda ,n=0,1,\dots ,s.$$

Thus, using the Formula (16), we obtain

$$\frac{1}{2\pi i}\underset{\vartheta \left(B\right)}{\int }{\phi }_1\left(\lambda \right)e^{-\phi \left(\lambda \right)t}B{(I-\lambda B)}^{-1}fd\lambda ={\phi }_1\left(W\right)u\left(t\right).$$

Consider a principal part of the Laurent series. Using the Formula (16), we get for values $n\in \mathbb{N}$

$$\frac{1}{2\pi i}\underset{\vartheta \left(B\right)}{\int }{\lambda }^{-n}{e}^{-\phi \left(\lambda \right)t}B{(I-\lambda B)}^{-1}fd\lambda =B^{n}u\left(t\right).$$

Not that by virtue of a character of the convergence of the series principal part, we have

$${∥\sum _{n=1}^{\infty }{c}_{-n}{e}^{-\phi \left(\lambda \right)t}{(I-\lambda B)}^{-1}{B}^{n+1}f∥}_{\mathfrak{H}}\le C{\parallel f\parallel }_{\mathfrak{H}}\sum _{n=1}^{\infty }\left|c_{-n}\right|·{∥B∥}_{\mathfrak{H}}^{n+1}<\infty ,\lambda \in \vartheta \left(B\right).$$

Therefore

$$\underset{{\vartheta }_j\left(B\right)}{\int }{\phi }_2\left(\lambda \right)e^{-\phi \left(\lambda \right)t}B{(I-\lambda B)}^{-1}fd\lambda =\sum _{n=1}^{\infty }{c}_{-n}\underset{{\vartheta }_j\left(B\right)}{\int }{e}^{-\phi \left(\lambda \right)t}{(I-\lambda B)}^{-1}{B}^{n+1}fd\lambda ,j\in \mathbb{N},$$

where

$${\vartheta }_j\left(B\right):=\left\{\lambda :\left|\lambda \right|=r>0,\left|\arg \lambda \right|\le \theta +\varsigma \right\}\cup \left\{\lambda :r<\left|\lambda \right|<r_j,r_j\uparrow \infty ,\left|\arg \lambda \right|=\theta +\varsigma \right\}.$$

Analogously to (19), we can easily get

$$e^{-\mathrm{Re}\phi \left(\lambda \right)t}\le e^{-C\left|\phi \right(\lambda \left)\right|t}\le e^{-{C\left|\lambda \right|}^{s}t},\lambda \in \vartheta \left(B\right).$$

(20)

Applying this estimate, we obtain

$${∥\sum _{n=1}^{\infty }{c}_{-n}\underset{{\vartheta }_j\left(B\right)}{\int }{e}^{-\phi \left(\lambda \right)t}{(I-\lambda B)}^{-1}{B}^{n+1}fd\lambda ∥}_{\mathfrak{H}}\le {C\parallel f\parallel }_{\mathfrak{H}}\sum _{n=1}^{\infty }|c_{-n}|·\parallel B^{n+1}\parallel \underset{{\vartheta }_j\left(B\right)}{\int }{e}^{-{C\left|\lambda \right|}^{s}t}\left|d\lambda \right|$$

$$\le {C\parallel f\parallel }_{\mathfrak{H}}\sum _{n=1}^{\infty }|c_{-n}{|·\parallel B\parallel }^{n+1}\underset{\vartheta \left(B\right)}{\int }{e}^{-{C\left|\lambda \right|}^{s}t}\left|d\lambda \right|<\infty .$$

Note that the uniform convergence of the series on the left-hand side with respect to j follows from the latter estimate. Reformulating the well-known theorem of calculus on the absolutely convergent series in terms of the norm, we have

$$\frac{1}{2\pi i}\underset{\vartheta \left(B\right)}{\int }{\phi }_2\left(\lambda \right)e^{-\phi \left(\lambda \right)t}B{(I-\lambda B)}^{-1}fd\lambda =\frac{1}{2\pi i}\sum _{n=1}^{\infty }{c}_{-n}\underset{\vartheta \left(B\right)}{\int }{e}^{-\phi \left(\lambda \right)t}{(I-\lambda B)}^{-1}{B}^{n+1}fd\lambda ={\phi }_2\left(W\right)u\left(t\right).$$

(21)

Thus, we obtain the first relation (14). Let us establish the second relation (14). Using the Formula (15), we obtain

$$\frac{1}{2\pi i}\underset{\vartheta \left(B\right)}{\int }{\lambda }^n{e}^{-\phi \left(\lambda \right)t}B{(I-\lambda B)}^{-1}fd\lambda =I_{1n}\left(t\right)+I_{2n}\left(t\right),n\in {\mathbb{Z}}^{-}\cup \{0,1,\dots ,s\},$$

where

$$I_{1n}\left(t\right):=\frac{1}{2\pi i}\underset{\vartheta \left(B\right)}{\int }{e}^{-\phi \left(\lambda \right)t}{\lambda }^{-2}{(I-\lambda B)}^{-1}{W}^{n+1}fd\lambda ,I_{2n}\left(t\right):=0,n=-2,$$

$$I_{2n}\left(t\right):=\left\{\begin{array}{c}\hfill -\sum _{k=-2}^{n-1}{\beta }_k\left(t\right)B^{k-n+1}f,n>-2,\\ \hfill \sum _{k=-3}^n{\beta }_k\left(t\right)B^{k-n+1}f,n\le -3\end{array}\right..$$

Using the proved above fact ${\beta }_k\left(t\right)=0,$ we have $I_{2n}\left(t\right)=0.$ Since in consequence of Lemma 1, inequality (20) for arbitrary $j\in \mathbb{N},f\in \mathrm{D}\left(W^s\right),$ we have

$$e^{-\phi \left(\lambda \right)t}{\lambda }^{-2}{(I-\lambda B)}^{-1}{W}^{n+1}f\to {\lambda }^{-2}{(I-\lambda B)}^{-1}{W}^{n+1}f,t\to +0,\lambda \in {\vartheta }_j\left(B\right),$$

where convergence is uniform with respect to the variable $\lambda ,$ the improper integral $I_{1n}\left(t\right)$ is uniformly convergent with respect to the variable $t,$ then we get

$$I_{1n}\left(t\right)\to \frac{1}{2\pi i}\underset{\vartheta \left(B\right)}{\int }{\lambda }^{-2}{(I-\lambda B)}^{-1}{W}^{n+1}fd\lambda ,t\to +0,$$

Note that the last integral can be calculated as a residue, we have

$$\begin{gathered} \frac{1}{2\pi i}\underset{\vartheta \left(B\right)}{\int }{\lambda }^{-2}{(I-\lambda B)}^{-1}{W}^{n+1}fd\lambda =\underset{\lambda \to 0}{lim}\frac{d{(I-\lambda B)}^{-1}}{d\lambda }{W}^{n+1}f=W^{n}f, \\ n\in {\mathbb{Z}}^{-}\cup \{0,1,\dots ,s\}. \end{gathered}$$

(22)

It is obvious that using this formula, we obtain the following relation

$$\underset{t\to +0}{lim}\frac{1}{2\pi i}\underset{\vartheta \left(B\right)}{\int }{\phi }_1\left(\lambda \right)e^{-\phi \left(\lambda \right)t}B{(I-\lambda B)}^{-1}fd\lambda =\sum _{n=0}^s{c}_n{W}^{n}f,f\in \mathrm{D}\left(W^s\right).$$

Consider a principal part of the Laurent series. The following reasonings are analogous to the above, we get

$$∥\sum _{n=1}^{\infty }{c}_{-n}\underset{\vartheta \left(B\right)}{\int }{e}^{-\phi \left(\lambda \right)t}{\lambda }^{-2}{(I-\lambda B)}^{-1}{B}^{n-1}fd\lambda ∥\le {C\parallel f\parallel }_{\mathfrak{H}}\sum _{n=1}^{\infty }|c_{-n}|·\parallel B^{n-1}\parallel \underset{\vartheta \left(B\right)}{\int }{\left|\lambda \right|}^{-2}{e}^{-{C\left|\lambda \right|}^{s}t}\left|d\lambda \right|$$

$$\le {C\parallel f\parallel }_{\mathfrak{H}}\sum _{n=1}^{\infty }|c_{-n}{|·\parallel B\parallel }^{n-1}\underset{\vartheta \left(B\right)}{\int }{\left|\lambda \right|}^{-2}\left|d\lambda \right|<\infty .$$

It gives us the uniform convergence of the series with respect to t at the left-hand side of the last relation. Using the analog of the well-known theorem of calculus on the absolutely convergent series, we have

$$\sum _{n=1}^{\infty }{c}_{-n}\underset{\vartheta \left(B\right)}{\int }{e}^{-\phi \left(\lambda \right)t}{\lambda }^{-2}{(I-\lambda B)}^{-1}{B}^{n-1}fd\lambda \to \sum _{n=1}^{\infty }{c}_{-n}\underset{\vartheta \left(B\right)}{\int }{\lambda }^{-2}{(I-\lambda B)}^{-1}{B}^{n-1}fd\lambda ,t\to +0.$$

Taking into account (3), (21), we get

$$\underset{t\to +0}{lim}\frac{1}{2\pi i}\underset{\vartheta \left(B\right)}{\int }{\phi }_2\left(\lambda \right)e^{-\lambda t}B{(I-\lambda B)}^{-1}fd\lambda =\sum _{n=1}^{\infty }{c}_{-n}{B}^{n}f,f\in \mathfrak{H}.$$

It is clear that the second relation (14) holds. The proof is complete. □

3.1. Existence and Uniqueness Theorems

This paragraph is the climax of the paper, here we represent a theorem that formed the beginning of marvelous research based on the Abell-Lidskii method. The attempt to consider an operator function in the second term was made in the paper [2], where we consider a case that is not so difficult since the corresponding function is of the power type. In contrast, in this paper, we consider a more complicated case, a function that compels us to involve a principally different method of study. The existence and uniqueness theorem given below is based on one of the number of theorems presented in [1].

In accordance with the ordinary approach [2], we consider a Hilbert space $\mathfrak{H}$ consists of element-functions $u:{\mathbb{R}}_{+}\to \mathfrak{H},u:=u\left(t\right),t\ge 0$ and we assume that if u belongs to $\mathfrak{H}$ then the fact holds for all values of the variable $t.$ We understand such operations as differentiation and integration in the generalized sense that is based on the topological properties of the Hilbert space $\mathfrak{H}.$ For instance, the derivative is understood as a limit in the sense of the norm, etc., more detailed information can be found in [1,27]. Let us study a Cauchy problem

$$\frac{du}{dt}+\phi \left(W\right)u=0,u\left(0\right)=h\in \mathrm{D}\left(W\right),$$

(23)

in this case, when the operator $\phi \left(W\right)$ is accretive we assume that $h\in \mathfrak{H}.$

Theorem 1.

Suppose B is a compact operator, $\mathsf{\Theta }\left(B\right)\subset {\mathfrak{L}}_0\left(\theta \right),\theta <\pi /2,B\in {\tilde{\mathfrak{S}}}_{\rho },$ moreover in the case $B\in {\tilde{\mathfrak{S}}}_{\rho }\backslash {\mathfrak{S}}_{\rho }$ the additional condition holds

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

(24)

the function φ is satisfied the conditions of Lemma 5, the following condition holds $s>\rho .$ Then a sequence of natural numbers ${\left\{N_{\nu }\right\}}_0^{\infty }$ can be chosen so that there exists a solution to the Cauchy problem (23) in the form

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

(25)

where

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

(26)

Moreover, the existing solution is unique if the operator $\phi \left(W\right)$ is accretive.

Proof. 

Firstly, let us establish relation (26). Consider a contour $\vartheta \left(B\right).$ Having fixed $R>0,0<\kappa <1,$ so that $R(1-\kappa )=r,$ consider a monotonically increasing sequence ${\left\{R_{\nu }\right\}}_0^{\infty },R_{\nu }=R{(1-\kappa )}^{-\nu +1}.$ Using Lemma 2, we get

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

where the function $\gamma \left(r\right)$ is defined in Lemma 2,

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

Note that in accordance with Lemma 3 [5] the following relation holds

$$\sum _{i=1}^{\infty }{\lambda }_i^{\frac{\rho +\epsilon }{(m+1)}}\left(\tilde{B}\right)\le \sum _{i=1}^{\infty }{s}_i^{\rho +\epsilon }\left(B\right)<\infty ,\epsilon >0,$$

(27)

where $\tilde{B}:={\left(B^{\ast m+1}{A}^{m+1}\right)}^{1/2}.$ It is clear that $\tilde{B}\in {\tilde{\mathfrak{S}}}_{\upsilon },\upsilon \le \rho /(m+1).$ Denote by ${\vartheta }_{\nu }$ a bound of the intersection of the ring ${\tilde{R}}_{\nu }<\left|\lambda \right|<{\tilde{R}}_{\nu +1}$ with the interior of the contour $\vartheta \left(B\right),$ denote by $N_{\nu }$ a number of poles being contained in the set $\mathrm{int}\vartheta \left(B\right)\cap \{\lambda :r<|\lambda |<{\tilde{R}}_{\nu }\}.$ In accordance with Lemma 3, we get

$$\frac{1}{2\pi i}\underset{{\vartheta }_{\nu }}{\oint }{e}^{-\phi \left(\lambda \right)t}B{(I-\lambda B)}^{-1}hd\lambda =\sum _{q=N_{\nu }+1}^{N_{\nu +1}}\sum _{\xi =1}^{m\left(q\right)}\sum _{i=0}^{k\left(q_{\xi }\right)}{e}_{q_{\xi }+i}{c}_{q_{\xi }+i}\left(t\right),h\in \mathfrak{H}.$$

(28)

Let us estimate the above integral, for this purpose split the contour ${\vartheta }_{\nu }$ on terms ${\tilde{\vartheta }}_{\nu }:=\{\lambda :|\lambda |={\tilde{R}}_{\nu },\left|\arg \lambda \right|\le \theta +\varsigma \},{\tilde{\vartheta }}_{\nu +1},{\vartheta }_{{\nu }_{+}}:=\{\lambda :{\tilde{R}}_{\nu }<\left|\lambda \right|<{\tilde{R}}_{\nu +1},\arg \lambda =\theta +\varsigma \},{\vartheta }_{{\nu }_{-}}:=\{\lambda :{\tilde{R}}_{\nu }<\left|\lambda \right|<{\tilde{R}}_{\nu +1},\arg \lambda =-\theta -\varsigma \}.$ Applying Lemma 2, relation (19), we get

$$J_{\nu }:={∥\underset{{\tilde{\vartheta }}_{\nu }}{\int }{e}^{-\phi \left(\lambda \right)t}B{(I-\lambda B)}^{-1}hd\lambda ∥}_{\mathfrak{H}}\le \underset{{\tilde{\vartheta }}_{\nu }}{\int }{e}^{-t\mathrm{Re}\phi \left(\lambda \right)}{∥B{(I-\lambda B)}^{-1}h∥}_{\mathfrak{H}}\left|d\lambda \right|$$

$$\le e^{{\gamma \left(\right|\lambda \left|\right)\left|\lambda \right|}^{\rho }}{\left|\lambda \right|}^{m+1}C{e}^{-{C\left|\lambda \right|}^{s}t}\underset{-\theta -\varsigma }{\overset{\theta +\varsigma }{\int }}d\arg \lambda ,\left|\lambda \right|={\tilde{R}}_{\nu }.$$

Thus, we get $J_{\nu }\le C{e}^{{\gamma \left(\right|\lambda \left|\right)\left|\lambda \right|}^{\rho }-C{\left|\lambda \right|}^{s}t}{\left|\lambda \right|}^{m+1},$ where $m=\left[\rho \right],\left|\lambda \right|={\tilde{R}}_{\nu }.$ Let us show that for a fixed t and a sufficiently large $\left|\lambda \right|,$ we have ${\gamma \left(\right|\lambda \left|\right)\left|\lambda \right|}^{\rho }-C{\left|\lambda \right|}^{s}t<0.$ It follows directly from Lemma 2 [1], we should consider (27), in the case when $B\in {\mathfrak{S}}_{\rho }$ as well as in the case $B\in {\tilde{\mathfrak{S}}}_{\rho }\backslash {\mathfrak{S}}_{\rho }$ but here we must involve the additional condition (24). Therefore

$$\sum _{\nu =0}^{\infty }{J}_{\nu }<\infty .$$

Using the analogous estimates, applying Lemma 1, we get

$$J_{\nu }^{+}:={∥\underset{{\vartheta }_{{\nu }_{+}}}{\int }{e}^{-\phi \left(\lambda \right)t}B{(I-\lambda B)}^{-1}hd\lambda ∥}_{\mathfrak{H}}\le {C\parallel h\parallel }_{\mathfrak{H}}·C\underset{R_{\nu }}{\overset{R_{\nu +1}}{\int }}{e}^{-Ct\mathrm{Re}\phi \left(\lambda \right)}\left|d\lambda \right|$$

$$\le C{e}^{-tC{R}_{\nu }^m}\underset{R_{\nu }}{\overset{R_{\nu +1}}{\int }}\left|d\lambda \right|=C{e}^{-tC{R}_{\nu }^m}\{R_{\nu +1}-R_{\nu }\}.$$

$$J_{\nu }^{-}:={∥\underset{{\vartheta }_{{\nu }_{-}}}{\int }{e}^{-\phi \left(\lambda \right)t}B{(I-\lambda B)}^{-1}hd\lambda ∥}_{\mathfrak{H}}\le C{e}^{-tC{R}_{\nu }^m}\underset{R_{\nu }}{\overset{R_{\nu +1}}{\int }}\left|d\lambda \right|=C{e}^{-tC{R}_{\nu }^m}\{R_{\nu +1}-R_{\nu }\}.$$

The obtained results allow us to claim (the proof is left to the reader) that

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

Using the Formula (28), the given above decomposition of the contour ${\vartheta }_{\nu },$ we obtain the relation (26). Let us establish (25), for this purpose, we should note that in accordance with relation (28), the properties of the contour integral, we have

$$\frac{1}{2\pi i}\underset{{\vartheta }_{{\tilde{R}}_p}\left(B\right)}{\oint }{e}^{-\phi \left(\lambda \right)t}B{(I-\lambda B)}^{-1}hd\lambda =\sum _{\nu =0}^{p-1}\sum _{q=N_{\nu }+1}^{N_{\nu +1}}\sum _{\xi =1}^{m\left(q\right)}\sum _{i=0}^{k\left(q_{\xi }\right)}{e}_{q_{\xi }+i}{c}_{q_{\xi }+i}\left(t\right),h\in \mathfrak{H},p\in \mathbb{N},$$

where the contour ${\vartheta }_{{\tilde{R}}_p}\left(B\right)$ is defined in Lemma 5. Using the proved above fact $J_{\nu }\to 0,\nu \to \infty ,$ we can easily get

$$\frac{1}{2\pi i}\underset{{\vartheta }_{{\tilde{R}}_p}\left(B\right)}{\oint }{e}^{-\phi \left(\lambda \right)t}B{(I-\lambda B)}^{-1}hd\lambda \to \frac{1}{2\pi i}\underset{\vartheta \left(B\right)}{\oint }{e}^{-\phi \left(\lambda \right)t}B{(I-\lambda B)}^{-1}hd\lambda ,p\to \infty .$$

The latter relation gives us the desired result (25). Let us show that $u\left(t\right)$ is a solution of the problem (23). Applying Lemma 5, we get

$$\phi \left(W\right)u\left(t\right)=\frac{1}{2\pi i}\underset{\vartheta \left(B\right)}{\int }\phi \left(\lambda \right)e^{-\phi \left(\lambda \right)t}B{(I-\lambda B)}^{-1}hd\lambda .$$

Now, we need to establish the following relation

$$\frac{du}{dt}=-\frac{1}{2\pi i}\underset{\vartheta \left(B\right)}{\int }\phi \left(\lambda \right)e^{-\phi \left(\lambda \right)t}B{(I-\lambda B)}^{-1}hd\lambda ,h\in \mathfrak{H},$$

(29)

i.e., we can use a differentiation operation under the integral. For this purpose, let us prove that for an arbitrary ${\vartheta }_j\left(B\right)$ (the definition is given in Lemma 5) there exists a limit

$$\frac{e^{-\phi \left(\lambda \right)\Delta t}-1}{\Delta t}{e}^{-\phi \left(\lambda \right)t}B{(I-\lambda B)}^{-1}h\stackrel{\mathfrak{H}}{⟶}-\phi \left(\lambda \right)e^{-\phi \left(\lambda \right)t}B{(I-\lambda B)}^{-1}h,\Delta t\to 0,$$

(30)

where convergence is uniform with respect to $\lambda \in {\vartheta }_j\left(B\right).$ Applying Lemma 1, we get

$${∥\frac{e^{-\phi \left(\lambda \right)\Delta t}-1}{\Delta t}{e}^{-\phi \left(\lambda \right)t}B{(I-\lambda B)}^{-1}h+\phi \left(\lambda \right)e^{-\phi \left(\lambda \right)t}B{(I-\lambda B)}^{-1}h∥}_{\mathfrak{H}}$$

$$\le C\left|\frac{e^{-\phi \left(\lambda \right)\Delta t}-1}{\Delta t}+\phi \left(\lambda \right)\right|\underset{\lambda \in {\vartheta }_j\left(B\right)}{max}{e}^{-\mathrm{Re}\phi \left(\lambda \right)t}.$$

It is clear that

$$\frac{e^{-\phi \left(\lambda \right)\Delta t}-1}{\Delta t}\to -\phi \left(\lambda \right),\Delta t\to 0,$$

where convergence, in accordance with the Heine-Cantor theorem, is uniform with respect to $\lambda \in {\vartheta }_j\left(B\right).$ Thus, we obtain (30). Using decomposition on the Taylor series, applying (20), we get

$${∥\frac{e^{-\phi \left(\lambda \right)\Delta t}-1}{\Delta t}{e}^{-\phi \left(\lambda \right)t}∥}_{\mathfrak{H}}\le \left|\phi \left(\lambda \right)\right|e^{\left|\phi \right(\lambda )\Delta t|}{e}^{-\mathrm{Re}\phi \left(\lambda \right)t}\le \left|\phi \left(\lambda \right)\right|e^{(\Delta t-Ct)\left|\phi \right(\lambda \left)\right|}$$

$$\le \left|\phi \left(\lambda \right)\right|e^{{(\Delta t-Ct)C\left|\lambda \right|}^s},\lambda \in \vartheta \left(B\right).$$

Thus applying the latter estimate, Lemma 1, for a sufficiently small value $\Delta t,$ we get

$${∥\underset{\vartheta \left(B\right)}{\int }\frac{e^{-\phi \left(\lambda \right)\Delta t}-1}{\Delta t}{e}^{-\phi \left(\lambda \right)t}B{(I-\lambda B)}^{-1}hd\lambda ∥}_{\mathfrak{H}}\le {C\parallel h\parallel }_{\mathfrak{H}}\underset{\vartheta \left(B\right)}{\int }{e}^{-{C\left|\lambda \right|}^s}{\left|\lambda \right|}^s\left|d\lambda \right|.$$

The function under the integral in the second term of the last relation guarantees that the improper integral at the left-hand side is uniformly convergent with respect to $\Delta t.$ These facts give us an opportunity to claim that the relation (29) holds. Here, we should explain that this conclusion is based upon the generalization of the well-known theorem of calculus. In its own turn, it follows easily from the theorem on the connection between the simultaneous limit and the repeated limit. We left a complete investigation of the matter to the reader, having noted that the scheme of the reasonings is absolutely the same in comparison with ordinary calculus. Thus, we obtain the fact that u is a solution of Equation (23).

Now, we want to show that the initial condition holds in the following sense $u\left(t\right)\underset{}{\overset{\mathfrak{H}}{\to }}h,t\to +0.$ It is obvious in the case $h\in \mathrm{D}\left(W\right),$ for in this case we can apply Lemma 4 and obtain the desired result, i.e., we can put $u\left(0\right)=h.$ Assume that h is an arbitrary element of the Hilbert space $\mathfrak{H}$ and let us apply the accretive property of the operator $\phi \left(W\right).$ In accordance with the above, for a fixed value of $t,$ we can understand a correspondence between $u\left(t\right)$ and h as an operator $S_t:\mathfrak{H}\to \mathfrak{H}.$ Let us prove that $\parallel S_t{\parallel }_{\mathfrak{H}\to \mathfrak{H}}\le 1,t>0.$ Firstly, assume that $h\in \mathrm{D}\left(W\right).$ Let us multiply both sides of the relation (23) on u in the sense of the inner product, we get ${\left(u_t^{\prime },u\right)}_{\mathfrak{H}}+{\left(\phi \left(W\right)u,u\right)}_{\mathfrak{H}}=0.$ Consider a real part of the last relation, we have $\mathrm{Re}{\left(u_t^{\prime },u\right)}_{\mathfrak{H}}+\mathrm{Re}{(\phi \left(W\right)u,u)}_{\mathfrak{H}}={\left(u_t^{\prime },u\right)}_{\mathfrak{H}}/2+{\left(u,u_t^{\prime }\right)}_{\mathfrak{H}}/2+\mathrm{Re}{(\phi \left(W\right)u,u)}_{\mathfrak{H}}.$ Therefore ${\left({\parallel u\left(t\right)\parallel }_{\mathfrak{H}}^2\right)}_t^{\prime }=-2\mathrm{Re}{(\phi \left(W\right)u,u)}_{\mathfrak{H}}\le 0.$ Integrating both sides from zero to $\tau >0,$ we get ${\parallel u\left(\tau \right)\parallel }_{\mathfrak{H}}^2-{\parallel u\left(0\right)\parallel }_{\mathfrak{H}}^2\le 0.$ The last relation can be rewritten in the form $\parallel S_t{h\parallel }_{\mathfrak{H}}\le {\parallel h\parallel }_{\mathfrak{H}},h\in \mathrm{D}\left(W\right).$ Since $\mathrm{D}\left(W\right)$ is a dense set in $\mathfrak{H},$ we obviously obtain the desired result, i.e., $\parallel S_t{\parallel }_{\mathfrak{H}\to \mathfrak{H}}\le 1.$ Now, having assumed that $h_n\underset{}{\overset{\mathfrak{H}}{\to }}h,n\to \infty ,\left\{h_n\right\}\subset \mathrm{D}\left(W\right),h\in \mathfrak{H},$ consider the following reasonings ${\parallel u\left(t\right)-h\parallel }_{\mathfrak{H}}=\parallel S_t{h-h\parallel }_{\mathfrak{H}}=\parallel S_{t}h-S_t{h}_n+S_t{h}_n-h_n+h_n{-h\parallel }_{\mathfrak{H}}\le \parallel S_t\parallel ·\parallel h-h_n{\parallel }_{\mathfrak{H}}+\parallel S_t{h}_n-h_n{\parallel }_{\mathfrak{H}}+{\parallel h_n-h\parallel }_{\mathfrak{H}}.$ Note that $S_t{h}_n\underset{}{\overset{\mathfrak{H}}{\to }}{h}_n,t\to +0.$ It is clear that if we choose n so that $\parallel h-h_n{\parallel }_{\mathfrak{H}}<\epsilon /3$ and after that choose t so that $\parallel S_t{h}_n-h_n{\parallel }_{\mathfrak{H}}<\epsilon /3,$ then we obtain ${\forall \epsilon >0,\exists \delta \left(\epsilon \right):\parallel u\left(t\right)-h\parallel }_{\mathfrak{H}}<\epsilon ,t<\delta .$ Thus, we can put $u\left(0\right)=h$ and claim that the initial condition holds in the case $h\in \mathfrak{H}.$ The uniqueness follows easily from the fact that $\phi \left(W\right)$ is accretive. In this case, repeating the previous reasonings, we come to ${\parallel \varphi \left(\tau \right)\parallel }_{\mathfrak{H}}^2\le {\parallel \varphi \left(0\right)\parallel }_{\mathfrak{H}}^2,$ where $\varphi$ is a sum of solutions $u_1$ and $u_2.$ Observe that by virtue of the initial conditions, we have $\varphi \left(0\right)=0.$ Therefore, the last inequality can hold only if $\varphi =0.$ We complete the proof. □

Remark 2.

Note that generally the existence and uniqueness Theorem 1 is based upon the Theorem 2 [1]. The corresponding analogs based upon Theorems 3 and 4 [1] can be obtained due to the same scheme and the proofs are not worth representing. At the same time, the mentioned analogs can be useful because of special conditions imposed upon the operator B such as ones formulated in terms of the operator order [1]. Here we should also appeal to the artificially constructed normal operator presented in [2].

3.2. Concrete Operators

It is remarkable that the made approach allows us to obtain a solution analytically for the evolution equation containing the second term—a function of an operator belonging to a sufficiently wide class of operators. Plenty of examples are presented in the paper [2] where such well-known operators as the Riesz potential, the Riemann-Liouville fractional differential operator, the Kipriyanov operator, the difference operator are considered. An interesting example can be also found in the paper [6]. A more general approach, implemented in the paper [14] allows us to build a transform of an operator belonging to the class of m-accretive operators. We should stress the significance of the last claim for the class containing the infinitesimal generator of a $C_0$ semigroup of contractions. In its own turn, fractional differential operators of the real order can be expressed in terms of the infinitesimal generator of the corresponding semigroup, which makes the offered generalization relevant (more detailed see [14]). Below, we present a rather abstract example for which the paper results can be applied. Consider a transform of an m-accretive operator J acting in $\mathfrak{H}$

$$Z_{G,F}^{\alpha }\left(J\right):=J^{\ast }GJ+F{J}^{\alpha },\alpha \in [0,1),$$

where symbols $G,F$ denote operators acting in $\mathfrak{H}.$ The Theorem 5 [14] gives us a tool to describe spectral properties of the transform $Z_{G,F}^{\alpha }\left(J\right).$ Particularly, we can establish the order of the transform and its belonging to the Schatten-von Neumann class of the convergence exponent by virtue of the Theorem 3 [14]. Thus, having known the index of the Schatten-von Neumann class of the convergence exponent, we can apply Theorem 1 to the transform.

4. Conclusions

In this paper, we invented a technique to study evolution equations with the second term a function of the non-selfadjoint unbounded operator. We remark that the main issue of the paper is an application of the spectral theorem to the special class of non-selfadjoint operators and in a natural way, we come to the definition of a function of the unbounded non-selfadjoint operator. From this point of view, the main highlights of this paper are propositions analogous to the spectral theorem. We can perceive them as an introduction or a way of reformulating the main principles of the spectral theorem based upon the peculiarities of the convergence in the Abell-Lidsky sense. In this regard, the main obstacle that appears is how to define an analog of a spectral family or decomposition of the identical operator.

However, as the main result, we have obtained an approach allowing us to principally broaden conditions imposed upon the second term 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 second term being presented by an operator function of a differential operator with a fractional derivative in final terms, are covered by the invented abstract method. In connection with these various types of fractional integro-differential operators can be considered. In additional, we can consider a class of the artificially constructed normal operators for which the clarification of the Lidskii results relevantly works. Apparently, the further step in the theoretical study may be to consider an entire function that generates the operator function, note that a prerequisite of the prospective result is given by Lemma 4. In this case, some difficulties may appear to relate to the propositions analogous to Lemma 5 and Theorem 1. In this regard, we can point out a technique presented in Chapter II of the monograph [23] that gives us more accurate asymptotic formulas. It is clear that having obtained the desired result for a class of entire functions, we can make significant progress in the direction since then we are able to broaden the rather primitive conditions imposed upon the regular part of the Laurent series. Having been inspired by the above ideas, we hope that the concept will have further development.