Factorizations along commutative subspace lattices

Abstract

A theorem of D. R. Larson on the factorization of positive-definite operators along complete, countable nests in a Hilbert space is generalized to the case of commutative subspace lattices: We characterize those selfadjoint, positive-definite operators which can be factored A^* A where A and A⁻¹ leave invariant the subspaces in a given countable, complete, commutative subspace lattice. Applications to inner-outer factorization theory and factorization of a positive-definite operator with respect an uncountable commutative subspace lattice are given. These results have applications to function theory on the polydisk, nonanticipative representations of Gaussian random fields, and multiparameter systems theory.