## Abstract and Applied Analysis

### On the Adjoint of a Strongly Continuous Semigroup

#### Abstract

Using some techniques from vector integration, we prove the weak measurability of the adjoint of strongly continuous semigroups which factor through Banach spaces without isomorphic copy of ${l}_{1}$; we also prove the strong continuity away from zero of the adjoint if the semigroup factors through Grothendieck spaces. These results are used, in particular, to characterize the space of strong continuity of ${\{{T}^{{\ast}{\ast}}(t)\}}_{t\geq{}0}$, which, in addition, is also characterized for abstract $L$- and $M$-spaces. As a corollary, it is proven that abstract $L$-spaces with no copy of ${l}_{1}$ are finite-dimensional.

#### Article information

Source
Abstr. Appl. Anal., Volume 2008 (2008), Article ID 651294, 11 pages.

Dates
First available in Project Euclid: 9 September 2008

https://projecteuclid.org/euclid.aaa/1220969148

Digital Object Identifier
doi:10.1155/2008/651294

Mathematical Reviews number (MathSciNet)
MR2393113

Zentralblatt MATH identifier
1165.47026

#### Citation

Bárcenas, Diómedes; Mármol, Luis Gerardo. On the Adjoint of a Strongly Continuous Semigroup. Abstr. Appl. Anal. 2008 (2008), Article ID 651294, 11 pages. doi:10.1155/2008/651294. https://projecteuclid.org/euclid.aaa/1220969148

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