Generalized Inner-Outer Factorizations in Non Commutative Hardy Algebras

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.