## Abstract

A rational function belongs to the Hardy space, H^{2}, of squaresummable 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 r ϵ H^{2}is particularly simple: The inner factor of 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 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.

Original language | English |
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Pages (from-to) | 6727-6749 |

Number of pages | 23 |

Journal | Transactions of the American Mathematical Society |

Volume | 374 |

Issue number | 9 |

DOIs | |

State | Published - 1 Jan 2021 |

## ASJC Scopus subject areas

- Mathematics (all)
- Applied Mathematics