Commuting Toeplitz operators with harmonic symbols

Powers and roots of Toeplitz operators

“…There are however some results which resemble those of [4]. In fact, Axler andCucković proved in [2] that the condition that one of (a), (b) or (c) be true is still necessary and sufficient when the two symbols φ and ψ are bounded harmonic functions on D. Moreover, with Rao [3], they proved that if φ is a bounded analytic function and if ψ is a bounded symbol such that T φ and T ψ commute, then ψ must be analytic too. When we consider arbitrary symbols, things are different.…”

A Characterization of M-Harmonicity

“…Their assumption is rather strong, but the proof is very simple and the result is good enough to apply to (the proof of) the main theorem of [2] which asserts that on the Bergman space, two Toeplitz operators with harmonic symbols commute only in the trivial cases. And then in 1993, Ahern, Flores, and Rudin [1] proved, by pure analytic methods, that on B n , an integrable function f with T 0 f = f has to be M-harmonic if and only if n ≤ 11 and a bounded function f with T 0 f = f is always M-harmonic.…”

“…, and much weaker than somewhat similar assumption imposed on the Lemma 2 of [2]. Here Rf denotes the radialization of f defined by…”

Commuting Separately Quasihomogeneous Small Hankel Operators on Pluriharmonic Bergman Space

“…References [4,5] obtained the Brown-Halmos type theorems for Toeplitz operators with harmonic symbols. Many subsequent works studied these problems for special symbol classes, such as harmonic symbols, radial symbols, or quasihomogeneous symbols; see [6][7][8][9], for example.…”