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

TL;DR

This paper investigates the spectral properties of constrained $K$-contractions on Hilbert spaces, using the support of their annihilators to identify joint spectra, with implications for functional models and spectral localization.

Contribution

It introduces the concept of support for annihilators of $K$-contractions and demonstrates how this support determines joint spectra in functional models, advancing spectral analysis techniques.

Findings

Support of annihilators determines joint spectra in functional models.

Localizable spectral points always exist within the support.

Small support allows effective detection of localizable points.

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.