Complex symmetric Toeplitz operators
The paper is devoted to study the complex symmetry for Toeplitz operators, with certain trigonometric symbols, over the Hardy space \(H^2\) of the unit circle \(\mathbb{T}\). A~bounded operator \(T\) on a Hilbert complex space is said to be complex symmetric (respectively, canonically symmetric) operator if there is a conjugation (respectively, canonical conjugation) \(C\) such that \(CTC=T^\star\). The general problem for characterizing which Toeplitz operators, even with monomial symbols, are complex symmetric is still open [\textit{K.-Y. Guo} and \textit{S. Zhu}, J. Oper. Theory 72, No. 2, 529--547 (2014; Zbl 1389.47042)]. For certain trigonometric symbols \(\varphi\) on \(\mathbb{T}\), the paper provides sufficient and necessary conditions for the Toeplitz operator \(T_\varphi\) to be complex symmetric on \(H^2\). The conditions are given in terms of the notion of unitarily equivalent transpose UET operators as well as the normality of the operator and the symmetry of the graph of the symbol viewed as a function on \([0,2\pi[\). Via Fourier series representation, the latter property is shown to be equivalent to the canonical symmetric property for Toeplitz with general bounded symbols on \(\mathbb{T}\) [\textit{R. Li} et al., J. Math. Anal. Appl. 489, No. 2, Article ID 124173, 11 p. (2020; Zbl 1481.47041)]. It is true that complex symmetric operators are UET operators, but the converse is not true [\textit{S. R. Garcia} and \textit{J. E. Tener}, J. Oper. Theory 68, No. 1, 179--203 (2012; Zbl 1274.15026)]. Thus, in order to the obtain the characterization, the authors mainly prove that, if \(T_\varphi\) is UET, then it is either normal or there is \(p\in\mathbb{T}\) with \(\varphi(e^{i\theta})=\varphi(pe^{-i\theta})\).
scientific article; zbMATH DE number 7345264
scientific article