The C*-algebra generated by irreducible Toeplitz and composition operators

TL;DR

This paper characterizes the C*-algebra generated by an irreducible Toeplitz operator with continuous symbol and certain composition operators on the Hardy space, highlighting differences from the algebra generated by shift and composition operators.

Contribution

It provides a description of the C*-algebra generated by specific Toeplitz and composition operators, including cases with automorphism symbols, and compares it to known operator algebras.

Findings

The algebra is described modulo compact operators.

For automorphism symbols, the algebra differs from the shift and composition algebra.

The structure of the algebra depends on the symbol and the inducing maps.

Abstract

We describe the C*-algebra generated by an irreducible Toeplitz operator $T_{\psi}$, with continuous symbol $\psi $ on the unit circle $\mathbb{T}$, and finitely many composition operators on the Hardy space $H^2$ induced by certain linear-fractional self-maps of the unit disc, modulo the ideal of compact operators $K(H^2)$. For composition operators with automorphism symbols, we show that this algebra is not isomorphic to the one generated by the shift and composition operators.