Localizable points in the support of a multiplier ideal and spectra of constrained operators

Abstract

A unitarily invariant, complete Nevanlinna–Pick kernel K on the unit ball determines a class of operators on Hilbert space called K-contractions. We study those K-contractions that are constrained, in the sense that they are annihilated by an ideal of multipliers. Our overarching goal is to identify various joint spectra of these constrained K-contractions through the vanishing locus of their annihilators. Our methods are based around a careful analysis of a subset of the ball associated to the annihilator, which we call its support. For the functional models, we show how this support completely determines several natural joint spectra. The picture is more complicated for general K-contractions, as their spectra can be properly contained in the support. Nevertheless, the “localizable” portion of the support always consists of spectral points. When the support is assumed to be small in an appropriate sense, we manage to effectively detect points of localizability.