Solving Operator Equations in Nest Algebras

Let X and Y be operators on Hilbert space, and let L be a nest of projections on the space. We consider the problem of finding an operator A in Alg L: such that A is Hilbert-Schmidt and such that AX = Y. A necessary and sufficient condition involving X,Y, and the projections in the lattice is found. We also indicate how the statements of the results can be modified so that the main theorem is true for any commutative subspace lattice Is. A number of authors have considered the equation Ax = y, where x and y represent given vectors in Hilbert space and the (bounded) operator A is to be found subject to certain criteria. For instance, suppose that JV is a nest of projections acting on the Hilbert space; what conditions on x and y guarantee the existence of an operator A E Alg N so that Ax = y? This question was discussed by Lance [7]. Several subsequent articles have generalized and expanded on Lance’s original result, including ones by Munch ([8]), Hopenwasser ([3], [4]), and the authors of this paper, in conjunction with Anoussis and Katsoulis ([l], [a], and [6]). Of particular interest for this article is Munch’s discussion of the problem of finding a Hilbert-Schmidt operator A in Alg N so that Ax = y, a problem motivated by general systems theory. Munch’s characterization and construction depend on the Arveson model, which represents commutative subspace lattices as lattices of increasing subsets of a partially ordered measure space. In this article, we adopt a point of view that proved useful in [6]; namely, we consider an operator equation AX = Y instead of the vector equation Ax = y. This change allows us to investigate both single-vector and multiple-vector interpolation simultaneously. At the same time, we want to use a “coordinate-free” approach, so we have avoided the Arveson model. We first consider the case in which N is a nest, and then we indicate how certain definitions can be changed so that the main theorem remains true for other CSL algebras. 483