2006/10/20 by Joseph A. Ball, Ball, Joseph A., Vladimir Bolotnikov +3 · 1 citation
Mathematics · #47A57 #Advanced Algebra and Geometry #Advanced Operator Algebra Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Holomorphic and Operator Theory #math.CA #msc:47A57
paper · pdf · doi:10.48550/arxiv.math/0610635
arxiv created 2006/10/20 · openalex publication_date 2006/10/20 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We introduce and study a Fock-space noncommutative analogue of reproducing kernel Hilbert spaces of de Branges-Rovnyak type. Results include: use of the de Branges-Rovnyak space \mathcal H(KS) as the state space for the unique (up to unitary equivalence) observable, coisometric transfer-function realization of the Schur-class multiplier S, realization-theoretic characterization of inner Schur-class multipliers, and a calculus for obtaining a realization for an inner multiplier with prescribed left zero-structure. In contrast with the parallel theory for the Arveson space on the unit ball \mathbb Bd ⊂ \mathbb Cd (which can be viewed as the symmetrized version of the Fock space used here), the results here are much more in line with the classical univariate case, with the extra ingredient of the existence of all results having both a ``left'' and a ``right'' version.