A certain rational function and operator theory on the symmetrized polydisc

Abstract

An n-tuple (S1,…,Sn−1,P) of commuting Hilbert space operators, for which the closed symmetrized polydiscΓn={(∑i=1nzi,∑1≤i<j≤nzizj,…,∏i=1nzi):|zi|≤1,i=1,…,n} is a spectral set, is called a Γn-contraction. Costara characterized a point in Γn (or in its interior Gn) via a rational function f˜s_ in the following way:s_=(s1,…,sn−1,p)∈Γn(or ∈Gn)⇔|f˜s_(z)|≤1(or <1),∀z∈D‾, where f˜s_ is the continuous extension of Costara's function fs_ defined on D byfs_(z)=n(−1)npzn−1+(n−1)(−1)n−1sn−1zn−2+…+(−s1)n−(n−1)s1z+…+(−1)n−1sn−1zn−1. In this paper, we investigate an operator theoretic analogue of this result. We prove that for a Γn-contraction only the forward implication holds and the converse holds for subnormal Γn-contractions. We provide a counterexample to show that the converse does not hold if we omit the subnormality condition. We obtain a few new characterizations of Γn-isometries and Γn-unitaries, which are analogues for Γn-contractions of the isometries and unitaries respectively in one variable operator theory. We show that (s1,…,sn−1,p)∈Γn if and only if (s1,…,sn−1,p,0,…,0︸k−times)∈Γn+k, but, the operator theoretic analogue of this result does not hold in general. We prove that if (S1,…,Sn−1,P) is a Γn-contraction then (S1,…,Sn−1,P,0,…,0︸k−times) is a Γn+k-contraction and construct a counterexample to show that the converse is not always true.