Factorization along commutative subspace lattices

A positive invertible operatorT is said to be factorable along a commutative subspace latticeL if there is an invertible operatorA inAlg L whose inverse is also inAlg L and such thatT=A*A. We investigate a number of conditions that are equivalent to factorability of a given operator along a latticeL. As a byproduct, we derive a condition that guarantees that the latticeT L, defined as {range(TE) E L} is commutative. Applications are suggested to the particular case of factoringL functions via analytic Toeplitz operators on the polydisc.