The $C^*$-algebra generated by irreducibleToeplitz and composition operators

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 specific automorphism-induced composition operators and certain types of irreducible Toeplitz operators, we show that the above $C^*$-al\-ge\-bra is not isomorphic to that generated by the shift and composition operators.