Irreducible approximation of Toeplitz operators

Abstract

This paper aims to study reducing subspaces of Toeplitz operators via an approximation approach. Halmos proved in 1968 that the set of irreducible operators on a separable Hilbert space is a dense $G_{\delta }$ set. We extend Halmos' result to the class $T_{C\left(T\right)}$ of Toeplitz operators with continuous symbols by proving that those irreducible ones constitute a dense $G_{\delta }$ subset of $T_{C\left(T\right)}$. In doing so, we give criteria to identify the irreducibility for trigonometric Toeplitz operators and matrices. As an application, we establish Halmos' approximation result for convex subsets of the $n\times n$ complex matrix algebra containing at least one irreducible element.