Toeplitz operators and Hilbert modules on the symmetrized polydisc

Abstract

When is the collection of [Formula: see text]-Toeplitz operators with respect to a tuple of commuting bounded operators [Formula: see text], which has the symmetrized polydisc as a spectral set, nontrivial? The answer is in terms of powers of [Formula: see text] as well as in terms of a unitary extension. En route, the Brown–Halmos relations are investigated. A commutant lifting theorem is established. Finally, we establish a general result connecting the [Formula: see text]-algebra generated by the commutant of [Formula: see text] and the commutant of its unitary extension [Formula: see text].