Toeplitz operators and Hilbert modules on the symmetrized polydisc

TL;DR

This paper investigates the structure of Toeplitz operators on the symmetrized polydisc, establishing conditions for non-triviality, exploring Brown-Halmos relations, and connecting commutants via a lifting theorem and $C^*$-algebra analysis.

Contribution

It provides new criteria for non-trivial $ extsf{S}$-Toeplitz operators, extends the Brown-Halmos relations, and links the $C^*$-algebras of commutants through a unitary extension.

Findings

Characterization of non-trivial $ extsf{S}$-Toeplitz operators.

Establishment of a commutant lifting theorem.

Connection between $C^*$-algebras of commutants and unitary extensions.

Abstract

When is the collection of $\mathsf S$-Toeplitz operators with respect to a tuple of commuting bounded operators $\mathsf S= (S_1, S_2, \ldots , S_{d-1}, P)$, which has the symmetrized polydisc as a spectral set, non-trivial? The answer is in terms of powers of $P$ as well as in terms of a unitary extension. En route, Brown-Halmos relations are investigated. A commutant lifting theorem is established. Finally, we establish a general result connecting the $C^*$-algebra generated by the commutant of $\mathsf S$ and the commutant of its unitary extension $\mathsf R$.