It is shown that hyper-reflexivity of a space of linear operators on a Hilbert space follows from a factorization property of linear functionals continuous in the weak operator topology. This provides new examples of hyper-reflexive algebras and new proofs for the hyper-reflexivity of the noncommutative disk algebras. 1998 Academic Press 1. INTRODUCTION Consider a complex Hilbert space H, and the algebra L(H) of bounded linear operators on H. A subspace M/L(H) is said to be reflexive if every operator T # L(H), with the property that Tx # [Mx] & for all x # H, necessarily belongs to M. This property can be formulated in a different way. Given vectors x, y # H, denote by x y the functional defined on L(H) by (T, x y) =(Tx, y), T # L(H).
We will also denote by [x y] M the restriction of x y to the subspace M. Then M is reflexive if, for every T ร M, there exist x, y # H satisfying [x y] M =0 and (Tx, y){0. A different formulation yet is given in terms of seminorms. Denote
The linear space M is said to be hyper-reflexive, or to satisfy a distance formula if there is a constant C>0 such that d M (T) Cr M (T ) for every T # L(H). The smallest constant C is called the hyper-reflexivity constant of M. These notions are usually introduced in terms of invariant subspaces when M is an algebra. The above notion of reflexivity for linear spaces was introduced in [12]. Distance formulas were first proved in [2] for nest algebras.