Complex symmetry of Toeplitz operators with continuous symbols

Abstract

In this note, we consider the question of when a Toeplitz operator on the Hardy–Hilbert space $H^2$ of the open unit disk $\mathbb {D}$ is complex symmetric, focusing on symbols $\phi :\mathbb {T}\rightarrow \mathbb {C}$ that are continuous on the unit circle $\mathbb {T}=\partial \mathbb {D}$. A closed curve $\phi$ is called nowhere winding if the winding number of $\phi$ is 0 about every point not in the range of $\phi$. It is then shown that if $T_\phi$ is complex symmetric, then $\phi$ must be nowhere winding. Hence if $\phi$ is a simple closed curve, then $T_\phi$ cannot be a complex symmetric operator. The spectrum and invertibility of complex symmetric Toeplitz operators with continuous symbols are then described. Finally, given any continuous curve $\gamma :[a,b]\rightarrow \mathbb {C}$, it is shown that there exists a complex symmetric Toeplitz operator with continuous symbol whose spectrum is precisely the range of $\gamma$.