Generalized Inner-Outer Factorization in non commutative Hardy Algebras

TL;DR

This paper extends the classical inner-outer factorization to non-commutative Hardy algebras associated with $W^*$-correspondences, providing new generalized factorizations for elements and their commutants.

Contribution

It introduces a generalized inner-outer factorization framework for non-commutative Hardy algebras, broadening previous special case results.

Findings

Constructed inner-outer factorizations for elements of $H^{ Infty}(E)$

Extended factorization results to the commutant of the algebra

Generalized classical results to a non-commutative setting

Abstract

Let $H^{\infty}(E)$ be a non commutative Hardy algebra, associated with a $W^*$-correspondence $E$. In this paper we construct factorizations of inner-outer type of the elements of $H^{\infty}(E)$ represented via the induced representation, and of the elements of its commutant. These factorizations generalize the classical inner-outer factorization of elements of $H^\infty(\mathbb{D})$. Our results also generalize some results that were obtained by several authors in some special cases.