scholarly journals Subalgebras of C*-algebras III: Multivariable operator theory

1998 ◽  
Vol 181 (2) ◽  
pp. 159-228 ◽  
Author(s):  
Whilliam Arveson
Keyword(s):  
Operator Theory ◽  
C Algebras ◽  
Multivariable Operator Theory

Noncommutative Function-Theoretic Operator Theory and Applications

2021 ◽  
Author(s):  
Joseph A. Ball ◽  
Vladimir Bolotnikov
Keyword(s):  
System Theory ◽  
Hardy Space ◽  
Operator Theory ◽  
Reproducing Kernel ◽  
Invariant Subspaces ◽  
Weighted Bergman Space ◽  
Characteristic Operator ◽  
Multivariable Operator Theory ◽  
Backward Shift ◽  
Space Function

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.

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