We prove the validity of the inverse formula of the Laplace transform for C 0 -semigroups in UMD-spaces in the strong sense and give an example which shows that the UMD-property is essential.
Given a family (e tAk ) t 0 (k # N) of commuting contraction semigroups, we investigate when the infinite product > k=1 e tAk converges and defines a C 0 -semigroup. A particular case is the heat semigroup in infinite dimension introduced by Cannarsa and Da Prato (J.
This paper investigates the admissibility of control and observation operators in UMD spaces. Necessary and/or sufficient conditions for unbounded control operators to be admissible and weakly admissible in the Salamon-Weiss sense are presented. This is illustrated by an example which shows that the UMD-property is essential. In particular, we get a direct proof of the known result of Driouich and and El-Mennaoui (Arch Math 72:56-63, 1999) on the validity of the inverse formula of the Laplace transform for C0-semigroups on UMD-spaces and in Hilbert spaces, as proved earlier by Yao (SIAM J Math Anal 26 (5): [1331][1332][1333][1334][1335][1336][1337][1338][1339][1340][1341] 1995). We outline how these results can be used to prove a partial validity of the inverse Laplace transform for semigroups in general Banach spaces. In particular, we obtain the result on the inverse Laplace transform due to Hille and Philllips (Am Math Soc Transl Ser 2, 1957).Mathematics Subject Classification (2010). Primary 99Z99; Secondary 00A00.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.