The spectral picture of Bergman–Toeplitz operators with harmonic polynomial symbols

Dechao Zheng (Center of Mathematics, Chongqing University, Chongqing, China; and Department of Mathematics, Vanderbilt University, Nashville, Tennessee, U.S.A.)

Abstract

This paper shows some new phenomenon in the spectral theory of Toeplitz operators on the Bergman space, which is considerably different from that of Toeplitz operators on the Hardy space. On the one hand, we prove that the spectrum of the Toeplitz operator with symbol $\overline{z}+p$ is always connected for every polynomial $p$ with degree less than $3$. On the other hand, we show that for each integer $k$ greater than $2$, there exists a polynomial $p$ of degree $k$ such that the spectrum of the Toeplitz operator with symbol $\overline{z}+p$ is a nonempty finite set. Then these results are applied to obtain a new class of non-hyponormal Toeplitz operators with bounded harmonic symbols on the Bergman space for which Weyl’s theorem holds.

Keywords

Bergman space, Toeplitz operator, harmonic polynomial, spectrum

2010 Mathematics Subject Classification

Primary 47B35. Secondary 47A10, 47B38.

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This work was partially supported by NSFC (Grant Nos.: 11701052, 11871157, 12371125).

The first author was supported by NNSF of China (12231005) and NSF of Shanghai (21ZR1404200).

The second author was supported by the Fundamental Research Funds for the Central Universities (Grant Nos.: 2020CDJQY-A039, 2020CDJLHSS-003).

Received 28 October 2021

Received revised 6 January 2023

Accepted 28 January 2023

Published 13 November 2023