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

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.