Noncommutative Function-Theoretic Operator Theory and Applications (Cambridge Tracts in Mathematics, Series Number 225)

This concise monograph explores how core ideas in Hardy space function theory and operator theory continue to be useful and informative in new settings, leading to new insights for noncommutative multivariable operator theory. Beginning with a review of the confluence of system theory ideas and reproducing kernel techniques, the book then covers representations of backward-shift-invariant subspaces in the Hardy space as ranges of observability operators, and representations for forward-shift-invariant subspaces via a Beurling–Lax representer equal to the transfer function of the linear system. This pair of backward-shift-invariant and forward-shift-invariant subspace form a generalized orthogonal decomposition of the ambient Hardy space. All this leads to the de Branges–Rovnyak model theory and characteristic operator function for a Hilbert space contraction operator. The chapters that follow generalize the system theory and reproducing kernel techniques to enable an extension of the ideas above to weighted Bergman space multivariable settings.

Joseph A. Ball is Professor Emeritus at Virginia Tech in Blacksburg, Virginia. He won Virginia Tech's Alumni Award for Research Excellence in 1997 and is a member of the 2019 class of Fellows of the American Mathematical Society. He is co-author of Interpolation of Rational Matrix Functions (1990).

Vladimir Bolotnikov is Professor of Mathematics at William & Mary in Williamsburg, Virginia. He has published over a hundred papers in operator and function theory.