Noncommutative rational Clark measures

We characterize the noncommutative Aleksandrov–Clark measures and the minimal realization formulas of contractive and, in particular, isometric noncommutative rational multipliers of the Fock space. Here, the full Fock space over $\mathbb {C} ^d$ is defined as the Hilbert space of square-summable po...

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Bibliographic Details
Published in:Canadian journal of mathematics 2023-10, Vol.75 (5), p.1393-1445
Main Authors: Jury, Michael T., Martin, Robert T.W., Shamovich, Eli
Format: Article
Language:English
Subjects:
Algebra
Control theory
Hilbert space
Mathematical analysis
Multipliers
Operators (mathematics)
Power series
Rational functions
Taylor series
Variables
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Summary:We characterize the noncommutative Aleksandrov–Clark measures and the minimal realization formulas of contractive and, in particular, isometric noncommutative rational multipliers of the Fock space. Here, the full Fock space over $\mathbb {C} ^d$ is defined as the Hilbert space of square-summable power series in several noncommuting (NC) formal variables, and we interpret this space as the noncommutative and multivariable analogue of the Hardy space of square-summable Taylor series in the complex unit disk. We further obtain analogues of several classical results in Aleksandrov–Clark measure theory for noncommutative and contractive rational multipliers. Noncommutative measures are defined as positive linear functionals on a certain self-adjoint subspace of the Cuntz–Toeplitz algebra, the unital $C^*$ -algebra generated by the left creation operators on the full Fock space. Our results demonstrate that there is a fundamental relationship between NC Hardy space theory, representation theory of the Cuntz–Toeplitz and Cuntz algebras, and the emerging field of noncommutative rational functions.
ISSN:0008-414X
1496-4279
DOI:10.4153/S0008414X22000384