FACTORIZATION AND REFLEXIVITY ON FOCK SPACES

The framework of the paper is that of the full Fock space F-2(H-n) and the Banach algebra F-infinity which can be viewed as non-commutative analogues of the Hardy spaces H-2 and H-infinity respectively. An inner-outer factorization for any element in F-2(H-n) as well as characterization of invertible elements in F-infinity are obtained. We also give a complete characterization of invariant subspaces for the left creation operators S-1,...,S-n of F-2(H-n). This enables us to show that every weakly (strongly) closed unital subalgebra of {phi(S-1,...,S-n):phi is an element of F-infinity} is reflexive, extending in this way the classical result of Sarason [S]. Some properties of inner and outer functions and many examples are also considered.