Hyponormal Toeplitz operators with non-harmonic Symbol acting on the Bergman space

Abstract

The Toeplitz operator acting on the Bergman space A 2 (D), with symbol ϕ is given by Tϕf = P (ϕf ), where P is the projection from L 2 (D) onto the Bergman space. We present some history on the study of hyponormal Toeplitz operators acting on A 2 (D), as well as give results for when ϕ is a non-harmonic polynomial. We include a first investigation of Putnam's inequality for hyponormal operators with non-analytic symbols. Particular attention is given to unusual hyponormality behavior that arises due to the extension of the class of allowed symbols.where σ(T ) denotes the spectrum of T (cf. [2]).We study the hyponormality of certain operators acting on the Bergman spaceLet ϕ ∈ L ∞ (D). The Toeplitz operator T ϕ is given bywhere P is the orthogonal projection from L 2 (D) onto A 2 (D).2010 Mathematics Subject Classification. 47B35, 47B20.

Hyponormal Toeplitz operators with non-harmonic algebraic symbol

“…then T ϕ is hyponormal. This condition is a substantial improvement to the conditions for hyponormality given in [6,Remark after Theorem 4].…”

“…In this section, we will answer the question (Q-III). In [6] Fleeman and Liaw consider non-harmonic polynomials and present the somewhat surprising example that T z+C|z| 2 is not hyponormal if |C| > 2 √ 2. They wonder for what values of C the operator T z+C|z| 2 is hyponormal.…”

“…In this section, we will list some generalizations of results from [6] that are related to question (Q-V). Many of these results share similar (or even identical) proofs to the corresponding results from [6], but we state them here with added generality for the sake of completeness.…”

“…A helpful formula that we will repeatedly use throughout this paper comes from [6] and is given by This formula will enable us to isolate the perturbations and understand their effect on hyponormality.…”

Normal Toeplitz Operators on the Bergman Space

“…Many authors in [1][2][3][4][5][6] studied intensively Normal operators and Toeplitz operators. It is a natural ask of when Toeplitz operator becomes normal.…”