Journal of Operator Theory

Volume 53, Issue 2, Spring 2005 pp. 315-329.

Strictly semi-transitive operator algebras

Authors: H.P. Rosenthal (1) and V.G. Troitsky (2)
Author institution: (1) Department of Mathematics, University of Texas, Austin, TX 78712, USA
(2) Department of Mathematics, University of Alberta, Edmonton, AB T6G 2G1, Canada

Summary: An algebra $\iA$ of operators on a Banach space $X$ is called strictly semi-transitive if for all non-zero $x,y\in X$ there exists an operator $A\in\iA$ such that $Ax=y$ or $Ay=x$. We show that if $\iA$ is norm-closed and strictly semi-transitive, then every $\iA$-invariant linear subspace is norm-closed. Moreover, $\Lat\iA$ is totally and well ordered by reverse inclusion. If $X$ is complex and $\iA$ is transitive and strictly semi-transitive, then $\iA$ is WOT-dense in $\iL(X)$. It is also shown that if $\iA$ is an operator algebra on a complex Banach space with no invariant operator ranges, then~$\iA$ is WOT-dense in $\iL(X)$. This generalizes a similar result for Hilbert spaces proved by Foiaș.

Keywords: Strictly semi-transitive algebra, transitive algebra, invariant subspace, operator range.

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