Spectra of $1/k$-Contractions via Characteristic Functions

Abstract

A unitarily invariant complete Nevanlinna--Pick (CNP) kernel $k$ on the Euclidean ball gives rise to a natural class of operator tuples on Hilbert spaces, known as $1/k$-contractions. We establish a lower estimate for the Taylor joint spectrum of $1/k$-contractions with finite defect in terms of the associated characteristic function. Under the additional assumption that $k$ satisfies the Corona property, this lower estimate coincides with an upper estimate due to Clouâtre--Timko [Adv. Math., 2023], yielding an exact characterization of the Taylor joint spectrum in terms of the characteristic function. As an application, we determine the Taylor joint spectrum of quotient modules of CNP spaces. A key ingredient in the proof is a Beurling--Lax--Halmos theorem for these spaces, established in terms of characteristic functions.