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Non-commutative rational functions in the full Fock space
A rational function belongs to the Hardy space, H^2, of square-summable power series if and only if it is bounded in the complex unit disk. Any such rational function is necessarily analytic in a disk of radius greater than one. The inner-outer factorization of a rational function \mathfrak {r} \in...
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Published in: | Transactions of the American Mathematical Society 2021-09, Vol.374 (9), p.6727-6749 |
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Main Authors: | , , |
Format: | Article |
Language: | English |
Citations: | Items that this one cites Items that cite this one |
Online Access: | Get full text |
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Summary: | A rational function belongs to the Hardy space, H^2, of square-summable power series if and only if it is bounded in the complex unit disk. Any such rational function is necessarily analytic in a disk of radius greater than one. The inner-outer factorization of a rational function \mathfrak {r} \in H^2 is particularly simple: The inner factor of \mathfrak {r} is a (finite) Blaschke product and (hence) both the inner and outer factors are again rational.
We extend these and other basic facts on rational functions in H^2 to the full Fock space over \mathbb {C} ^d, identified as the non-commutative (NC) Hardy space of square-summable power series in several NC variables. In particular, we characterize when an NC rational function belongs to the Fock space, we prove analogues of classical results for inner-outer factorizations of NC rational functions and NC polynomials, and we obtain spectral results for NC rational multipliers. |
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ISSN: | 0002-9947 1088-6850 |
DOI: | 10.1090/tran/8418 |