Cauchy Problem for an Abstract Evolution Equation of Fractional Order

Abstract

In this paper, we define an operator function as a series of operators corresponding to the Taylor series representing the function of the complex variable. In previous papers, we considered the case when a function has a decomposition in the Laurent series with the infinite principal part and finite regular part. Our central challenge is to improve this result having considered as a regular part an entire function satisfying the special condition of the growth regularity. As an application, we consider an opportunity to broaden the conditions imposed upon the second term not containing the time variable of the evolution equation in the abstract Hilbert space.

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

“…where ε is an arbitrary positive number. Bellow, we will show that the definition is correct and coincides with the one given in [22] as well as the information on the analytic functions with the growth regularity. Using simple reasonings involving properties of the resolvent, Cauchy integral formula e.t.c., we get…”

“…The passage to the limit when t tends to zero is justified by the same fact. Thus, we have obtained the equivalence of definitions, compare with the one given in [22]). It is clear that considering the set U t f, f ∈ N, t > 0, we can expand the domain of definition of the operator function ϕ at the same time the extension remains closed as one can easily see.…”

“…Note that condition (9) plays the distinguished role in the refinement of the Lidskii results [18] since it guaranties the equality of the convergence exponent and the order of summation in the Abell-Lidskii sense. It is rather reasonable to expect that we are highly motivated to produce a concrete example of the operator satisfying the condition (9) for if we found it then it would stress the significance and novelty of the papers [14,15,16,18,17,19,20,22]. Having been inspired by the idea, we can use the function considered in Example 1 as an indicator to find the desired operator function.…”

“…Here, we ought to explain that we managed to pass to the limit considering the contour integrals by virtue of the growth regularity of the function, a complete scheme of reasonings is represented in [20], [22]. Thus, the given above definition corresponds to the closure of the operator function considered in [22] (see Lemma 3) and defined on the subset of elements depended on the parameter i.e.…”

“…Most likely, the general theory created in the papers [15], [17] can be adopted to some operator composition but it is a tremendous work. Instead of that, in the paper [22] we succeed to find a suitable pair of Hilbert spaces what allows us to apply theoretical results.…”