Totally Abelian Toeplitz operators and geometric invariants associated with their symbol curves

Abstract: This paper mainly studies totally Abelian operators in the context of analytic Toeplitz operators on both the Hardy and Bergman space. When the symbol is a meromorphic function on $\mathbb{C}$, we establish the connection between totally Abelian property of these operators and and geometric properties of their symbol curves. It is found that winding numbers and multiplicities of self-intersection of symbol curves play an important role in this topic. Techniques of group theory, complex analysis, geometry and operator theory are intrinsic in this paper. As a byproduct, under a mild condition we provides an affirmative answer to a question raised in \cite{BDU,T1}, and also construct some examples to show that the answer is negative if the associated conditions are weakened.