Noncommutative Function-Theoretic Operator Theory and Applications

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.

• Presents a broad overview of the whole book in the context of the classical Hardy space function theory setting, providing readers with an easy reference point for comparing where the general setting still carries the classical structure, and where and how it departs significantly from the classical setting
• Includes remarks and examples throughout the text, helping readers understand the broader context of the material, how a result foreshadows a more general result to come, and how loose ends in a current result are resolved by a more sophisticated result to come in a later chapter
• Provides a Notes section at the end of each chapter to point readers to recent literature with results closely related to those in the book and possible new directions for future research
• Makes explicit connections with the work on Bergman-inner functions and the Bergman shift operator from the 1990s