Complex symmetric Toeplitz operators on the Hardy spaces and Bergman spaces

Abstract

In this paper, we first completely characterize the complex symmetric Toeplitz operators \(T_\varphi \) on the Hardy spaces \(H^2({\mathbb {D}})\) with conjugations \({\mathcal {C}}_p^{i,j}\) and \({\mathcal {C}}_n\). Next, we give a method to determine the coefficients of \(\varphi (z)\) when \(T_\varphi \) is complex symmetric on \(H^2({\mathbb {D}})\) with the conjugation \({\mathcal {C}}_\sigma \), which partially solves a problem raised by [2]. Finally, we consider the complex symmetric Toeplitz operators \(T_\varphi \) on the weighted Bergman spaces \(A^2({\mathbb {B}}_{n})\) and the pluriharmonic Bergman spaces \(b^2({\mathbb {B}}_{n})\) with conjugations \({\mathcal {C}}_V\), where V is a symmetric permutation matrix.