Reflexive Operator Algebra - Hyper-reflexivity

Hyper-reflexivity

Let be a weak*-closed operator algebra contained in B(H), the set of all bounded operators on a Hilbert space H and for T any operator in B(H), let

.

Observe that P is a projection involved in this supremum precisely if the range of P is an invariant subspace of .

The algebra is reflexive if and only if for every T in B(H):

.

We note that for any T in B(H) the following inequality is satisfied:

.

Here is the distance of T from the algebra, namely the smallest norm of an operator T-A where A runs over the algebra. We call hyperreflexive if there is a constant K such that for every operator T in B(H),

.

The smallest such K is called the distance constant for . A hyper-reflexive operator algebra is automatically reflexive.

In the case of a reflexive algebra of matrices with nonzero entries specified by a given pattern, the problem of finding the distance constant can be rephrased as a matrix-filling problem: if we fill the entries in the complement of the pattern with arbitrary entries, what choice of entries in the pattern gives the smallest operator norm?