When is the collection of $S$ -Toeplitz operators with respect to a tuple of commuting bounded operators $S = (S_{1}, S_{2}, \dots, S_{d - 1}, P)$ , 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 $R$ .

Communicated by Yasuyuki Kawahigashi

Keywords:

AMSC: 47A13, 47A20, 47B35, 46L07

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