On 2-Hyponormal Toeplitz Operators With Finite Rank Self-Commutators

Abstract

Suppose Tϕ is a 2-hyponormal Toeplitz operator whose selfcommutator has rank n ≥ 1. If Hφ ker [T * ϕ , Tϕ] contains a vector en in a canonical orthonormal basis {e k } k∈Z + of H 2 (T), then ϕ should be an analytic function of the form ϕ = qh, where q is a finite Blaschke product of degree at most n and h is an outer function.