Toeplitz operators and Hilbert modules on the symmetrized polydisc

When is the collection of S-Toeplitz operators with respect to a tuple of commuting bounded operators S = (S-1, S-2, ..., 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.