Transitivity and bundle shifts

Abstract

A subalgebra $A$ of the algebra $B(\mathcal{H})$ of bounded linear operators on a separable Hilbert space $\mathcal{H}$ is said to be catalytic if every transitive subalgebra $\mathcal{T}\subset B(\mathcal{H})$ containing it is strongly dense. We show that for a hypo-Dirichlet or logmodular algebra, $A=H^{\infty}(m)$ acting on a generalized Hardy space $H^{2}(m)$ for a representing measure $m$ that defines a reproducing kernel Hilbert space is catalytic. For the case of a nice finitely-connected domain, we show that the "holomorphic functions" of a bundle shift yields a catalytic algebra, thus generalize a result of Bercovici, Foias, Pearcy and the first author[7].