Localizable points in the support of a multiplier ideal and spectra of constrained operators (Q6039830)

Starting from the fact that the spectrum of a square matrix is given as the zero set of its minimal polynomial, infinite-dimensional multivariate analogues are established in this paper, using \(C_0\)-contractions theory. By Beurling's theorem it is known that the ideal \(\{f\in H^\infty(\mathbb D) : f(A)=0\}\) is generated by an inner function \(\theta\) and the spectrum of \(A\) satisfies the equality \(\sigma(A)=\sup (\theta)\), equality which is generalized here from an operator to finite tuple of operators and in the case of operators whose structure is encoded by a wider class of algebras of functions. The class of contractions is generalized to to the class of commuting \(K\)-contraction, where \(K\) is a reproducing kernel of a Hilbert space \(\mathcal H\), and the corresponding functional calculus is considered. After presenting the topic and necessary preliminaries on multivariate theory in the first two sections of the paper, in the third section the support of an ideal of multipliers is introduced and an analogue of equality \(\sigma(A)=\sup (\theta)\) is obtained (Theorem 3.8). Introducing the notion of localizability of an ideal, in the last section of the paper localizable points in the support are studied. Since for a regular unitary invariant complete Nevanlina-Pick space \(\mathcal H\) with kernel \(K\) the spectrum of an absolutely continuous commuting \(K\)-contraction \(T=(T_1,\dots,T_d)\) is included in the support of generalized annihilator of \(T\), under some conditions the converse inclusion is obtained. Various examples are given about strict inclusions on both sides, and cases when the equality is obtained.