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A dichotomy theorem for the adjoint of a semigroup of operators


Author: J. M. A. M. van Neerven
Journal: Proc. Amer. Math. Soc. 119 (1993), 765-774
MSC: Primary 47D03
DOI: https://doi.org/10.1090/S0002-9939-1993-1203994-6
MathSciNet review: 1203994
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Abstract: Let $ T(t)$ be a $ {C_0}$-semigroup of linear operators on a Banach space $ X$, and let $ {X^ \otimes }$, resp. $ {X^ \odot }$, denote the closed subspaces of $ {X^{\ast}}$ consisting of all functionals $ {x^{\ast}}$ such that the map $ t \mapsto {T^{\ast}}(t){x^{\ast}}$ is strongly continuous for $ t > 0$, resp. $ t \geqslant 0$.

Theorem. Every nonzero orbit of the quotient semigroup on $ {X^{\ast}}/{X^ \otimes }$ is nonseparably valued. In particular, orbits in $ {X^{\ast}}/{X^ \odot }$ are either zero for $ t > 0$ or nonseparable. It also follows that the quotient space $ {X^{\ast}}/{X^ \otimes }$ is either zero or nonseparable. If $ T(t)$ extends to a $ {C_0}$-group, then $ {X^{\ast}}/{X^ \odot }$ is either zero or nonseparable.

For the proofs we make a detailed study of the second adjoint of a $ {C_0}$-semigroup.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1993-1203994-6
Keywords: Adjoint semigroup, $ {C_0}$-semigroup, Baire-$ 1$ functional, Pettis integral, weakly Borel measurable
Article copyright: © Copyright 1993 American Mathematical Society

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