Summary: FACTORIZATION AND REFLEXIVITY ON FOCK SPACES
Alvaro Arias and Gelu Popescu
The framework of the paper is that of the full Fock space F2(Hn) and the Banach
algebra F1 which can be viewed as non-commutative analogues of the Hardy spaces H2
and H1 respectively.
An inner-outer factorization for any element in F2(Hn) as well as characterization
of invertible elements in F1 are obtained. We also give a complete characterization of
invariant subspaces for the left creation operators S1 Sn of F2(Hn). This enables us
to show that every weakly (strongly) closed unital subalgebra of f'(S1 Sn) : ' 2 F1g
is re exive, extending in this way the classical result of Sarason S]. Some properties of
inner and outer functions and many examples are also considered.
0. INTRODUCTION. Let Hp, 1 p 1 be the classical Hardy spaces on
the disk. H2 is a Hilbert space with orthonormal basis fzng1n=0 and it is well known that
if ' 2 H1, then k'k1 = kM'k, where M' : H2 ! H2 is the multiplication operator.
(1) k'k1 = sup fk'pk2 : kpk2 1 and p is a polynomial on zg:
The set of polynomials on z, P(z), determine the Hardy spaces: H2 is the closure
of P(z) in the Hilbert space with orthonormal basis fzng1n=0. Once we have H2, H1
consists of all ' 2 H2 such that (1) is nite.
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