Journal of Operator Theory

Volume 57, Issue 1, Winter 2007 pp. 19-34.

The behavior of the radical of the algebras generated by a semigroup of operators on Hilbert space

Authors: H.S. Mustafayev
Author institution: Yuzuncu Yil University, Faculty of Arts and Sciences, Department of Mathematics, Van, Turkey

Summary: Let

$\textbf{\textit{T}}=\{T(t)\}_{t\geqslant0}$ be a continuous semigroup of contractions on a Hilbert space. We define

$\textit{\textbf{A(T)}}$ as the closure of the set

$\{{\widehat{f}(\textit{\textbf{T}}):f\in L^1({\mathbb{R}_+})} \}$ with respect to the operator-norm topology, where

$\widehat{f}(\textit{\textbf{T}})=\int\limits_0^\infty {f(t )}T(t)\mathrm dt$ is the Laplace transform of

$f\in L^1({\mathbb{R}_+})$ with respect to the semigroup \textbf{\textit{T}}. Then,

$\textit{\textbf{A(T)}}$ is a commutative Banach algebra. In this paper, we obtain some connections between the radical of

$\textit{\textbf{A(T)}}$ and the set

$\{R\in \textbf{\textit{A(T)}}:T(t)R\to 0,\mbox{strongly or in}$

$\mbox{norm}, \mbox{as }t\to \infty\}$ . Similar problems for the algebras generated by a discrete semigroup

$\{T^{n}:n=0,1,2,\ldots \}$ is also discussed, where

$T$ is a contraction.

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