Blaschke-Singular-Outer factorization of free non-commutative functions

Abstract

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, H 2 , 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.

Non-commutative rational functions in the full Fock space

“…Proof. By Theorem 5.2, the inner factor, Θ of r is NC Blaschke, so that r is outer if and only if r is pointwise invertible in B d ℵ 0 by [13,Theorem 4.2]. In particular, r is NC outer if and only if r(rL) −1 ∈ H ∞ d for any 0 < r < 1, and this happens if and only if the radius of convergence of the Taylor series at 0 is at least 1.…”

“…As established in [13], any element, h, of Fock space has a unique NC Blaschke-Singular-Outer factorization, h = B • S • f where B ∈ H ∞ d is an NC Blashcke inner function, S ∈ H ∞ d is NC singular inner and f ∈ H 2 d is NC outer. Namely, as proven by Popescu and Davidson-Pitts, h = Θ • f has a unique NC inner-outer factorization, where Θ ∈ H ∞ d is inner, i.e.…”

“…cyclic for the right shifts. In [13] we proved that the inner factor, Θ can be further uniquely factored as the product of an NC Blaschke inner function, B, containing all the vanishing or zero information of h, and an NC singular inner function which is pointwise invertible in…”

“…In [13], the authors have refined this NC inner-outer factorization by extending the classical Blaschke-Singular-Outer factorization to the NC Hardy spaces H ∞ d and H 2 d . Namely, any NC inner Θ ∈ H ∞ d further factors uniquely as the product of an NC Blaschke inner and an NC singular inner.…”