Hyponormal Toeplitz operators with non-harmonic algebraic symbol

Abstract

Given a bounded function ϕ on the unit disk in the complex plane, we consider the operator T ϕ , defined on the Bergman space of the disk and given by T ϕ (f ) = P (ϕf ), where P denotes the projection to the Bergman space in L 2 (D, dA). We provide new necessary conditions on ϕ for T ϕ to be hyponormal, extending recent results of Fleeman and Liaw. One of our main results provides a necessary condition on the complex constant C for the operator T z n +C|z| s to be hyponormal. This condition is also sufficient if s ≥ 2n.

Hyponormal Toeplitz Operators on Weighted Bergman Space

“…is the numerator of the entry J (ν) n+k,k and hence the discussion before Theorem 2.1 implies each of these terms is strictly positive. That discussion also implies γ 2k+2n > γ 2 2k γ 2k−2n when k ≥ n. Thus we may reason as in [16] and conclude that T z n +C|z| s is hyponormal if and only if…”

“…Theorem 2.1 below will complete that result by providing necessary and sufficient conditions on the constant C for T ϕ acting on any A 2 ν (D) to be hyponormal. As a result, we will recover the aforementioned result from [16]. Our condition is stated in terms of the norm of a certain self-adjoint operator that happens to be a block Jacobi matrix.…”

“…Throughout this section, let us suppose that n ∈ N, s ∈ (0, ∞), and ν as in Section 1.1 are fixed. As in [5,16], we will use the formula…”

“…There is an extensive literature aimed at characterizing those symbols ϕ for which the corresponding operator T ϕ is hyponormal, much of which focuses on the special case of the classical Bergman space of the unit disk (see [1,3,5,6,7,8,9,10,11,15,16]). The specific symbol we will focus on is ϕ(z) = z n + C|z| s , where n ∈ N, C ∈ C, and s ∈ (0, ∞).…”

“…The specific symbol we will focus on is ϕ(z) = z n + C|z| s , where n ∈ N, C ∈ C, and s ∈ (0, ∞). The case n = 1 and s = 2 in the classical Bergman space was considered in [5] while a broader range of n and s was previously considered in [16], where it was shown that hyponormality of T ϕ acting on the classical Bergman space implies |C| ≤ n s and the converse holds if s ≥ 2n. Theorem 2.1 below will complete that result by providing necessary and sufficient conditions on the constant C for T ϕ acting on any A 2 ν (D) to be hyponormal.…”