Linear Equations in Subspaces of Operators

Given a subspace S of operators on a Hilbert space, and given two operators X and Y (not necessarily in S), when can we be certain that there is an operator A in S such that AX = Y ? If there is one, can we find some bound for its norm? These questions are the subject of a number of papers, some by the present authors, and mostly restricted to the case where S is a reflexive algebra. In this paper, we relate the broader question involving operator subspaces to the question about reflexive algebras, and we examine a new method of forming counterexamples, which simplifies certain constructions and answers an unresolved question. In particular, there is a simple set of conditions that are necessary for the existence of a solution in the reflexive algebra case; we show that — even in the case where the co-rank of X is one— these conditions are not in general sufficient. In a sequence of papers ( [2], [3], [4] ) the authors have studied the possibility of solving the operator equation AX = Y , where X and Y are given Hilbert space operators and A is an operator to be found. In those papers, the operator A is required to lie in the algebra associated with some commutative subspace lattice. Certain necessary conditions were derived in [3] for the existence of such an operator, and, in [4], we showed that those conditions are not sufficient. The construction of the counterexample in [4] was somewhat complicated, and the structure of the algebra studied there is obscure. In this paper we suggest a new way to construct counterexamples, which — on the one hand — connects the problems of finding unknown operators in CSL algebras with the problem of finding them in other operator subspaces, and — on the other — allows us to answer some unresolved questions about the sufficiency of the conditions derived in [3]. Throughout, H will represent a separable Hilbert space with inner product 〈·, ·〉; H will sometimes be finite-dimensional. A subspace is a closed linear manifold, and we will often identify the subspace with the orthogonal projection whose range it is. If L is a collection of projections (or subspaces) that is closed under the operations of meet and join, it is a lattice. If, in addition, it contains the zero and identity operators, and is closed in the strong operator topology, it is called a subspace lattice. If the projections in L all commute with each other, then L is a commutative subspace lattice, or CSL. The collection of operators that leave invariant all the projections in L is called Alg L, and it is easy to see that Alg L is a weakly closed algebra of operators. Standard terminology refers to the collection Alg L as a reflexive algebra, and — if L is a CSL — as a CSL algebra. Received by the editors April 22, 1998. 1991 Mathematics Subject Classification. Primary 47D25. c ©1999 American Mathematical Society