Editing Reflexive operator algebra (section)

== Hyper-reflexivity ==

Let <math>\mathcal{A}</math> 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

:<math>\beta(T,\mathcal{A})=\sup \left\{ \left\| P^\perp TP \right\| \ :\ P\mbox{ is a projection and } P^\perp \mathcal{A} P = (0) \right\} .</math>

Observe that ''P'' is a projection involved in this supremum precisely if the range of ''P'' is an invariant subspace of <math>\mathcal{A}</math>.

The algebra <math>\mathcal{A}</math> is reflexive if and only if for every ''T'' in ''B''(''H''):

:<math>\beta(T,\mathcal{A})=0 \mbox{ implies that } T \mbox{ is in } \mathcal{A} .</math>

We note that for any ''T'' in ''B(H)'' the following inequality is satisfied:

:<math>\beta(T,\mathcal{A})\le \mbox{dist}(T,\mathcal{A}) .</math>

Here <math>\mbox{dist}(T,\mathcal{A})</math> 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 <math>\mathcal{A}</math> '''hyperreflexive''' if there is a constant ''K'' such that for every operator ''T'' in ''B''(''H''),

:<math>\mbox{dist}(T,\mathcal{A})\le K \beta(T,\mathcal{A}) .</math>

The smallest such ''K'' is called the '''distance constant''' for <math>\mathcal{A}</math>. 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?