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~Foia{\c{s}}.

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