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 $$\gamma $$ .