Global convoluted semigroups

Abstract

Global exponentially bounded convoluted semigroups in Banach spaces are systematically treated with the help of Laplace transform. A perturbation theorem in this context is proved and some characterizations of the introduced class of analytic convoluted semigroups are obtained. Illustrative examples of generators of convoluted semigroups, including differential operators, are presented.

Hyperfunction semigroups

“…In general, the second condition does not hold for exponentially bounded functions, cf. [3, Theorem 1.11.1] and [31]. Following analysis in [10] and [29, Theorem 2.7.1, Theorem 2.7.2], in our context, we can give the following statements:…”

Differential and Analytical Properties of Semigroups of Operators

“…In Theorem 2.19, Corollary 2.20 and Theorem 2.22, we analyze differential properties of perturbed fractionally integrated C-semigroups and (C-)distribution semigroups, and continue the researches of Da Prato and Mosco [26,27] and Fujiwara [38] concerning analytic distribution semigroups. We revisit the backwards heat equation and answer affirmatively to the problem proposed in [63]. Section 3 is devoted to the study of corresponding properties of ultradistribution semigroups.…”

“…The relationship between exponential (UDSG)'s of the Beurling class and exponentially bounded convoluted semigroups has been recently discussed in [63]. Notice that the preceding theorem enables one to transfer the assertion of [63, Theorem 3.10] to Roumieu case as well as to remove the density assumption from the formulation of [63, Theorem 3.10(i)].…”

“…Suppose now t ∈ (0, 1). Then it can be simply verified [63,67] that, for every s ∈ [0, 1] and f ∈ D(G(δ t )), (G(δ t )f )(s) = 0, 0 ≤ s ≤ t, f (s − t) , 1 ≥ s > t.…”

“…In the present paper, we deal with the notion of differentiability and analyticity of (local) convoluted C-semigroups [6,19,20,50,63,64,66], C-distribution semigroups and (ultra-)distribution semigroups (cf. [15,16,18,34,51,56,62,65,[72][73][74][75]78,84,86,107,109] and references therein) and continue the researches of regularity properties of strongly continuous, integrated and 500 M. Kostić I E O T C-regularized semigroups [4,5,[8][9][10]22,23,28,33,36,[44][45][46]66,93,94,101,104,[112][113][114][115][116].…”