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.

‘Noncommutative Function-Theoretic Operator Theory and Applications by Ball and Bolotnikov is a comprehensive monograph by acknowledged experts in the fields of operator theory and function theory. It gives an account of a very active area of modern research, to which the authors themselves have been major contributors. The significant themes of the book include reproducing kernel Hilbert spaces (notably weighted Bergman spaces), Beurling-Lax theorems, and systems-theoretic ideas expressed in operator-theoretic terms. The work as a whole is presented in a multivariable noncommutative context, and thus extends classical work on Hardy-space function theory and related operator theory.'

Jonathan Partington - University of Leeds