Toeplitz operators and Hilbert modules on the symmetrized polydisc

Abstract

When is the collection of S-Toeplitz operators with respect to a tuple of commuting bounded operators S=(S1,S2,…,Sd−1,P), which has the symmetrized polydisc as a spectral set, nontrivial? The answer is in terms of powers of P 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 C∗-algebra generated by the commutant of S and the commutant of its unitary extension R.