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.

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

“…(21)In accordance with (15), we have(ln 1+1/α x) ′ s −1 m (B) = o(m −1/α ), =⇒ ln r n(r) r ρ → 0, r → ∞ ,Applying(17), we haven B m+1 (r m+1 ) ≤ (m + 1)n B (r), hence ln r n B m+1 (r m+1 ) r α ≤ (m + 1) ln r n B (r) r α ,what gives us the desired result, i.e. we compleat the proof of the first relation(20).…”

“…Substituting δ −1 ν , we have In accordance with Lemma 6 [18], we can claim that on each ray ζ containing the point zero and not belonging to the sector L 0 (θ) as well as the real axis, we have It obvious, that the same estimate can be obtained for J − ν . Therefore, the second and the third relations (20) hold. The proof is complete.…”

“…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.…”

Schatten Index of the Sectorial Operator via the Real Component of Its Inverse

“…The abstract approach to the Cauchy problem for the fractional evolution equation is a classic one [12,13]. In its framework, the application of results connected with the basis property covers many problems in the theory of evolution equations [1,10,[14][15][16]. In its general statement, the problem appeals to many applied ones, and we can produce a number of papers dealing with differential equations which can be studied by the abstract methods [17][18][19][20][21][22].…”