Convoluted C-groups

Abstract

We introduce and systematically analyze the class of convoluted C-groups unifying the well known classes of fractionally integrated groups and C-regularized groups. We relate convoluted C-groups to analytic convoluted C-semigroups and present illustrative examples of differential operators which generate exponentially bounded convoluted groups.

Distribution groups

“…Definition 2.4. [26] Let and be closed linear operators, ∈ (0, ∞] and > 0. A strongly continuous operator family ( ( )) ∈(− , ) is called a (local, if < ∞) -times integrated group generated by if:…”

“…On the other hand, global integrated groups were introduced and investigated by El-Mennaoui in his doctoral dissertation [13]. We refer the reader to [3]- [6], [12]- [13], [16], [18]- [20], [26] and, especially, to the paper [33] where Miana analyzed global -times integrated groups and smooth distribution groups in the framework of fractional calculus. It is also meaningful to accent that Keyantuo [20] briefly considered an abstract Laplacian in (R )-type spaces and proved several relations between exponentially bounded integrated cosine functions and global integrated groups.…”

“…For further information, see [20,Theorem 1.2, Propositions 2.1-2.2, Theorem 2.6 and Proposition 4.2]. The class of (local) convoluted -groups extending the well known classes of integrated groups and regularized groups has been recently introduced in [26].…”

Notes on analytic convoluted C-semigroups

“…In Theorem 2.1, we introduce the condition (H 1 ) which holds in the case of fractionally integrated C-semigroups. In order to better explain the importance of this condition in our investigation, let us recall that the set ℘(S K ) consisted of all subgenerators of a (local) convoluted C-semigroup (S K (t)) t∈[0,τ ) need not be finite ( [8], [10], [13]) and that, equipped with corresponding algebraic operations, ℘(S K ) becomes a complete lattice whose partially ordering coincides with the usual set inclusion; furthermore, ℘(S K ) is totally ordered iff card(℘(S K )) 2 ( [10], [13]), and in the case card(℘(S K )) < ∞, one can prove that ℘(S K ) is a Boolean, which implies card(℘(S K )) = 2 n for some n ∈ N 0 . In fact, the main objective in Theorem 2.1(i) is to establish the spectral characterizations of the integral generator of an analytic convoluted C-semigroup (S K (t)) t 0 as well as to show that such characterizations still hold for an arbitrary subgenerator of (S K (t)) t 0 as long as the condition (H 1 ) holds.…”