Hyponormal Toeplitz operators and extremal problems of Hardy spaces

Abstract

Abstract.The symbols of hyponormal Toeplitz operators are completely described and those are also studied, being related with the extremal problems of Hardy spaces. Moreover, we discuss Halmos's question about a subnormal Toeplitz operator when the self-commutator is finite rank.

Hyponormality and spectra of Toeplitz operators

“…(The existence of a such a Blaschke product in E(ϕ) is predicted by Theorem 10 of [18].) The discussion above views the solution c 0 , .…”

“…Cowen's theorem can be stated as follows (see [18,Lemma 1]): a Toeplitz operator T ϕ is hyponormal if and only if the subset E(ϕ) of H ∞ is nonempty. Suppose that ϕ is the trigonometric polynomial ϕ(e iθ ) = N n=−N a n e inθ , where …”

“…-by studying the function ϕ itself. In a recent paper [18] of T. Nakazi and K. Takahashi, Cowen's method is carried out to obtain substantial new information about hyponormal Toeplitz operators and their symbols. In the present paper we study the hyponormality of T ϕ in the cases where ϕ is a trigonometric polynomial ϕ(e iθ ) = N −m a n e inθ ; the goal here is to find conditions on the coefficients a n that are necessary and sufficient for T ϕ to be hyponormal.…”

A Note on Hyponormal Toeplitz Operators

“…The assertions (i) -(vi) were shown from [4,6,7,10,11,12,13,16]. For the assertion (vii), suppose that there exists a function…”

Hyponormality of block Toeplitz operators

“…An equivalent condition in [Nakazi and Takahashi 1993] is that ϕ − kϕ ∈ H ∞ . A generalization of Cowen's result for the hyponormality of generalized Toeplitz operators was obtained in [Gu 1994].…”