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A unitarily invariant, complete Nevanlinna--Pick kernel $K$ on the unit balldetermines a class of operators on Hilbert space called $K$-contractions. Westudy those $K$-contractions that are constrained, in the sense that they areannihilated by an ideal of multipliers. Our overarching goal is to identifyvarious joint spectra of these constrained $K$-contractions through thevanishing locus of their annihilators. Our methods are based around a carefulanalysis of a subset of the ball associated to the annihilator, which we callits support. For the functional models, we show how this support completelydetermines several natural joint spectra. The picture is more complicated forgeneral $K$-contractions, as their spectra can be properly contained in thesupport. Nevertheless, the "localizable" portion of the support always consistsof spectral points. When the support is assumed to be small in an appropriatesense, we manage to effectively detect points of localizability.

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