Narrow Results

We introduce a notion of hyperbolicity and parabolicity for a holomorphic self-map $f:{\mathrm{\Delta }}^N\to {\mathrm{\Delta }}^N$ of the polydisc which does not admit fixed points in ${\mathrm{\Delta }}^N$. We generalize to the polydisc two classical one-variable results: we solve the Valiron equation for a hyperbolic $f$ and the Abel equation for a parabolic nonzero-step $f$. This is done by studying the canonical Kobayashi hyperbolic semi-model of $f$ and by obtaining a normal form for the automorphisms of the polydisc. In the case of the Valiron equation, we also describe the space of all solutions.

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 $R$.