Non-commutative rational functions in the full fock space

Michael T. Jury, Robert T.W. Martin, Eli Shamovich

Research output: Contribution to journalArticlepeer-review

6 Scopus citations


A rational function belongs to the Hardy space, H2, 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 H2is 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 H2to the full Fock space over Cd, 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 languageEnglish
Pages (from-to)6727-6749
Number of pages23
JournalTransactions of the American Mathematical Society
Issue number9
StatePublished - 1 Jan 2021

ASJC Scopus subject areas

  • Mathematics (all)
  • Applied Mathematics


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