Zeros of Holomorphic Functions in Commuting and Non-commuting Variables as Spectral Data

TL;DR

The paper develops a unified spectral-data framework for zero sets of holomorphic functions in commuting and non-commuting variables by leveraging unitary realizations. Zeros in commutative domains (the Euclidean ball and polydisk) are characterized via row and diagonal eigenvalues of the realization operator $D^*$, while in NC settings zeros align with NC $Q$-eigenvalues, with boundary zeros governed by NC approximate point spectra and boundary-value phenomena. This approach extends classical one-variable factorization ideas (Blaschke products, Smirnov factorization) to several variables and NC domains, linking zero sets to operator-theoretic data and revealing structural parallels across commutative and NC contexts. The results include explicit determinantal representations for zeros of rational inner functions and a detailed boundary analysis, highlighting the interplay between function theory, realization theory, and spectral theory with potential impact on interpolation and factorization in multivariable and NC settings.

Abstract

We characterize the zero sets of functions in the Schur--Agler class over the unit polydisk as well as functions in the unit ball of the multiplier algebra of the Drury--Arveson space via operators associated with a unitary realization formula for these functions. To this end, new notions of `eigenvalues' for tuples of operators are introduced, where the eigenvalues depend on the operator space structure of the ambient domain. Several examples showcasing the properties of these eigenvalues and the zero sets of rational inner functions in the Schur--Agler class are also presented. We further generalize this result to a large class of non-commuting (NC) holomorphic functions whose ambient domain is given by the unit ball of a matrix of linear polynomials. This includes the NC counterparts of the unit polydisk and the Euclidean unit ball. We also show for functions in the Schur--Agler class over NC matrix unit balls that their zeros along the topological boundary are contained in an appropriately defined `approximate point spectrum' of the associated realization operator, and so are points along the Shilov boundary where the boundary values are not isometric/coisometric. This, in-turn, provides an identical result for the commutative case.

Theorems & Definitions (35)

• Theorem 1.1
• Definition 2.1
• Theorem A
• Definition 2.2
• Theorem B
• Definition 2.3
• Theorem C
• Theorem D
• Theorem E
• Remark 3.1