Invertibility and spectral properties of dual Toeplitz operators

Abstract

In this paper, we consider dual Toeplitz operators with continuous symbols and bounded harmonic symbols on the orthogonal complement of the Bergman space over the open unit disk. We construct a dual Toeplitz operator with continuous symbol such that its spectrum is disconnected. On the other hand, we show that the spectra of dual Toeplitz operators with certain class of bounded harmonic symbols are connected, which leads to some partial answers to the open question posed by Stroethoff and Zheng [18] in 2002. We obtain a complete characterization for the hyponormality of dual Toeplitz operators with bounded harmonic symbols. Moreover, we establish some sufficient conditions that are convenient to check for dual Toeplitz operators with continuous symbols and bounded harmonic symbols to be invertible, respectively.