Reflexive operator algebra

In functional analysis, a reflexive operator algebra "A" is an operator algebra that has enough invariant subspaces to characterize it. Formally, "A" is reflexive if it is equal to the algebra of bounded operators which leave invariant each subspace left invariant by every operator in "A".
This should not be confused with a reflexive space.
Examples.
Nest algebras are examples of reflexive operator algebras. In finite dimensions, these are simply algebras of all matrices of a given size whose nonzero entries lie in an upper-triangular pattern.
In fact if we fix any pattern of entries in an "n" by "n" matrix containing the diagonal, then the set of all "n" by "n" matrices whose nonzero entries lie in this pattern forms a reflexive algebra.
An example of an algebra which is "not" reflexive is the set of 2 × 2 matrices
This algebra is smaller than the Nest algebra
but has the same invariant subspaces, so it is not reflexive.
If "T" is a fixed "n" by "n" matrix then the set of all polynomials in "T" and the identity operator forms a unital operator algebra. A theorem of Deddens and Fillmore states that this algebra is reflexive if and only if the largest two blocks in the Jordan normal form of "T" differ in size by at most one. For example, the algebra
which is equal to the set of all polynomials in
and the identity is reflexive.
Hyper-reflexivity.
Let formula_5 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 formula_5.
The algebra formula_5 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 formula_11 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 formula_5 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 formula_5. 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?