Laplace transforms of vector-valued functions with growth ω and semigroups of operators

Abstract

For an arbitrary continuous, increasing function ω: [0, ∞) → C of finite exponential type, we characterize Laplace-Stieltjes transforms of Banach-spacevalued functions that are O(ω) , and use this to establish a Hille-Yosida type theorem for strongly continuous semigroups that are O(ω) . Corollaries include characterizing generators of strongly continuous semigroups that are O(exp(dt 1 d )) , for d > 1 , in addition to the already known examples of exponential or polynomial growth. 1,2

Strongly continuous integrated cosine operator functions with growth ω

“…Suppose g : (r 0 , ∞) → C, where r 0 is the constant defined in (2). Then the following statements are equivalent:…”

Hille–Yosida type theorems for -times integrated semigroups

“…Other important contributions on this topic have been made by Kellermann and Hieber in [3], Thieme in [4], deLaubenfels in [5,6], among others. O(1 + t k ) once integrated semigroups were investigated in [8,9]. It should be pointed out that Mijatović, Pilipović and Vajzović established Hille-Yosida type theorems for α-times integrated semigroups (α ∈ R + ) in [7], Xiao and Liang established approximation theorems for α-times integrated semigroups under weaker conditions in [14].…”

“…In [8], the authors introduced the Laplace-Stieltjes transform of O(ω(t)) vector-valued functions in a unified way, and obtained Hille-Yosida type theorems for O(ω(t)) once integrated semigroups.…”

“…In particular, we will establish a Hille-Yosida type theorem for n-times integrated semigroups with non-quasi-analytic growth (Theorem 4.6) including O exp dt 1 d (d > 1) growth (Theorem 4.7). In Section 2, as a preparation, we collect some basic facts from [8,12].…”

“…In order to deal with the n-times integrated Laplace transform of vector-valued functions in a more general setting, we borrowed from [8] a function ω(t) : [0, ∞) → C that satisfies the following conditions:…”