Approximate Factorization in Generalized Hardy Spaces

Abstract

In this paper we establish a connection between the approximate factorization property appearing in the theory of dual algebras and the spectral inclusion property for a class of Toeplitz operators on Hilbert spaces of vector valued square integrable functions. As an application, it follows that a wide range of dual algebras of subnormal Toeplitz operators on various Hardy spaces associated to function algebras have property (A1 (1)). It is also proved that the dual algebra generated by a spherical isometry (with a possibly infinite number of components) has the same property. One particular application is given to the existence of unimodular functions sitting in cyclic invariant subspaces of weak* Dirichlet algebras. Moreover, by this method we provide a unified approach to several Toeplitz spectral inclusion theorems. Mathematics Subject Classification (2000). Primary 47L45, 47B35; Secondary 47B20, 46E40, 46J10, 46J15, 46E30.

A Factorization Theorem for Multiplier Algebras of Reproducing Kernel Hilbert Spaces

“…where M g is the multiplication operator on L 2 (ν, D) induced by g. For more details on this see [15,Lemma 4.1]. Let H = Γ(H), and for any x ∈ H denote x = Γ(x).…”