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 spaceAuthors: 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.
Keywords: Hilbert space, continuous (discrete) semigroup, Banach algebra, radical.
Contents Full-Text PDF