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 inAlgL whose inverse is also inAlgL 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 latticeTL, 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.