Differential and Analytical Properties of Semigroups of Operators

Kostić, Marko

doi:10.1007/s00020-010-1797-4

Cited by 9 publications

(9 citation statements)

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“…The main objective in Theorems 2.20-2.24 is to enquire into the basic differential properties of a, k -regularized C-resolvent families. Theorem 2.20 25 . SupposeA is a closed linear operator, k t and a t satisfy (P1), r ≥ −1, and there exists ω ≥ max 0, abs k , abs a such that, for every z ∈ {λ ∈ C : Re λ > ω, k λ / 0}, we have that the operator I − a z A is injective and that Rang C ⊆ Rang I − a z A .…”

Section: 37mentioning

confidence: 99%

(a, k)‐Regularized C‐Resolvent Families: Regularity and Local Properties

Kostić

2009

Abstract and Applied Analysis

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We introduce the class of local a, k -regularized C-resolvent families and discuss its basic structural properties. In particular, our analysis covers subjects like regularity, perturbations, duality, spectral properties and subordination principles. We apply our results in the study of the backwards fractional diffusion-wave equation and provide several illustrative examples of differentiable a, k -regularized C-resolvent families. Abstract and Applied AnalysisThroughout this paper E denotes a nontrivial complex Banach space, L E denotes the space of bounded linear operators from E into E, E * denotes the dual space of E, and A denotes a closed linear operator acting on E. The range and the resolvent set of Aare denoted by Rang A and ρ A , respectively; D A denotes the Banach space D A equipped with the graph norm. From now on, we assume that L E C is an injective operator which satisfies CA ⊆ AC and employ the convolution like mapping * which is given by f * g t : t 0 f t − s g s ds. Recall, the C-resolvent set of A, denoted by ρ C A , is defined to be the set of all complex numbers λ satisfying that the operator λ − A is injective and that Rang C ⊆ Rang λ − A . Let us recall that a linear subspace Y ⊆ D A is called a core for A if Y is dense in D A with respect to the graph norm. Henceforth we identify a closed linear operator A with its graph G A ; given two closed linear operators A and B on E,We mainly use the following conditions. a, k -Regularized C-Resolvent FamiliesWe start with the following definition.Definition 2.1. Let 0 < τ ≤ ∞, k ∈ C 0, τ , k / 0, and let a ∈ L 1 loc 0, τ , a / 0. A strongly continuous operator family R t t∈ 0,τ is called a local, if τ < ∞ a, k -regularized Cresolvent family having A as a subgeneratorif and only if the following holds:In the case τ ∞, R t t≥0 is said to be exponentially bounded if, additionally, there exist M > 0 and ω ≥ 0 such that R t ≤ Me ωt , t ≥ 0; R t t∈ 0,τ is said to be nondegenerate if the condition R t x 0, t ∈ 0, τ implies x 0.From now on, we consider only nondegenerate a, k -regularized C-resolvent families. Notice that R t t∈ 0,τ is nondegenerate provided that k 0 / 0 or that H5 holds for a subgenerator A of R t t∈ 0,τ . Abstract and Applied Analysis 3In the case k t t α /Γ α 1 , where α > 0, and Γ · denotes the Gamma function, it is also said that R t t∈ 0,τ is an α-times integrated a, C -resolvent family; in such a way, we unify the notion of local α-times integrated C-semigroups a t ≡ 1 and cosine functions a t ≡ t 1, 13, 14 . Furthermore, in the case k t : t 0 K s ds, t ∈ 0, τ , where K ∈ L 1 loc 0, τ and K / 0, we obtain the unification concept for local K-convoluted Csemigroups and cosine functions 15 . In the case k t ≡ 1, R t t∈ 0,τ is said to be a local a, C -regularized resolvent family with a subgenerator A cf. also 16 for the definition which does not include the condition ii of Definition 2.1 .Designate by ℘ R the set which consists of all subgenerators of R t t∈ 0,τ . Then the following holds.

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Section: 37mentioning

confidence: 99%

(a, k)‐Regularized C‐Resolvent Families: Regularity and Local Properties

Kostić

2009

Abstract and Applied Analysis

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“…It would take too long to go into details concerning stability of certain differential properties ( [40,41]) under bounded commuting perturbations described in Theorem 5.…”

Section: Theoremmentioning

confidence: 99%

“…57Then ( ) = C, generates a tempered ultradistribution semigroup of ( ! )-class, and cannot be the generator of a distribution semigroup since is not stationary dense (see e.g., [53, Example 1.6] and [41]). If ∈ , ∈ [0, 1] and ∈ C,…”

Section: Theoremmentioning

confidence: 99%

“…)-class provided ≥ 2 , and ( ) generates an exponentially bounded, L −1 ( − 1/ )-convoluted group provided > | | −1/ / cos( /2 ). Let us also mention that the consideration given in example following [41,Corollary 3.8] enables one to construct important examples of (pseudo-)differential operators generating ultradistribution sines, and that the estimate 67…”

Section: Theoremmentioning

confidence: 99%

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Perturbation Theory for Abstract Volterra Equations

Kostić

2013

Abstract and Applied Analysis

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“…Ultradistribution semigroups in Banach spaces, with densely or non-densely defined generators, and abstract Beurling spaces have been analyzed in the papers of R. Beals [8]- [9], J. Chazarain [12], I. Ciorȃnescu [14], I. Ciorȃnescu, L. Zsido [17], P. R. Chernoff [18], H. A. Emami-Rad [24] and H. Komatsu [38] (cf. also [41], [45], [46]- [47], [53] and [60]). On the other hand, the study of distribution semigroups in locally convex spaces has been initiated by R. Shiraishi, Y. Hirata [76], T. Ushijima [79] and M. Ju Vuvunikjan [80].…”

Section: Introductionmentioning

confidence: 99%