A spectral dichotomy for commuting m-isometries with negative core operator

Abstract

Let T=(T1,T2) be a pair of commuting operators such that the Taylor spectrum σ(T) of T is contained in the closed unit bidisc D‾2 and the left spectrum of either of T1 or T2 is contained in the unit circle ∂D. If the core operator of T is negative, then we show that σ(T) is either contained in ∂D2 or equal to D‾2. This fact applies in particular to a commuting pair of m-isometries with negative core operator. Our method of proof relies on a strictly 2-variable fact about the topological boundary of the Taylor spectrum. We also present several applications of this dichotomy to the multivariate Fredholm theory.