Toeplitz operators and division by inner functions

Abstract

A subspace X of the Hardy space H1 is said to have the K-property if for any ψ ∈ H∞, the Toeplitz operator Tψ maps X into itself. This in turn implies that X also has the f-property. This means that h/I ∈ X whenever h ∈ X and I is an inner function with h/I ∈ H1. In this survey paper we present a list of subspaces of H1 that have or have not the f- or K-property, showing some of the different techniques and methods used in the subject.