TY - JOUR

T1 - Blaschke–singular–outer factorization of free non-commutative functions

AU - Jury, Michael T.

AU - Martin, Robert T.W.

AU - Shamovich, Eli

N1 - Funding Information:
First named author partially supported by NSF grant DMS-1900364 . Second author partially supported by NSERC grant 2020-05683 .
Publisher Copyright:
© 2021 Elsevier Inc.

PY - 2021/6/25

Y1 - 2021/6/25

N2 - By classical results of Herglotz and F. Riesz, any bounded analytic function in the complex unit disk has a unique inner–outer factorization. Here, a bounded analytic function is called inner or outer if multiplication by this function defines an isometry or has dense range, respectively, as a linear operator on the Hardy Space, H2, of analytic functions in the complex unit disk with square-summable Taylor series. This factorization can be further refined; any inner function θ decomposes uniquely as the product of a Blaschke inner function and a singular inner function, where the Blaschke inner contains all the vanishing information of θ, and the singular inner factor has no zeroes in the unit disk. We prove an exact analogue of this factorization in the context of the full Fock space, identified as the Non-commutative Hardy Space of analytic functions defined in a certain multi-variable non-commutative open unit ball.

AB - By classical results of Herglotz and F. Riesz, any bounded analytic function in the complex unit disk has a unique inner–outer factorization. Here, a bounded analytic function is called inner or outer if multiplication by this function defines an isometry or has dense range, respectively, as a linear operator on the Hardy Space, H2, of analytic functions in the complex unit disk with square-summable Taylor series. This factorization can be further refined; any inner function θ decomposes uniquely as the product of a Blaschke inner function and a singular inner function, where the Blaschke inner contains all the vanishing information of θ, and the singular inner factor has no zeroes in the unit disk. We prove an exact analogue of this factorization in the context of the full Fock space, identified as the Non-commutative Hardy Space of analytic functions defined in a certain multi-variable non-commutative open unit ball.

KW - Blaschke-singular-outer factorization

KW - Fock space

KW - Inner-outer factorization

KW - Non-commutative Hardy space

KW - Non-commutative analysis

UR - http://www.scopus.com/inward/record.url?scp=85103378554&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2021.107720

DO - 10.1016/j.aim.2021.107720

M3 - Article

AN - SCOPUS:85103378554

SN - 0001-8708

VL - 384

JO - Advances in Mathematics

JF - Advances in Mathematics

M1 - 107720

ER -