# 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 by 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
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?

## Examples

- Every finite-dimensional reflexive algebra is hyper-reflexive. However, there are examples of infinite-dimensional reflexive operator algebras which are not hyper-reflexive.
- The distance constant for a one-dimensional algebra is 1.
- Nest algebras are hyper-reflexive with distance constant 1.
- Many von Neumann algebras are hyper-reflexive, but it is not known if they all are.
- A type I von Neumann algebra is hyper-reflexive with distance constant at most 2.

## See also

- Invariant subspace
- subspace lattice
- reflexive subspace lattice
- nest algebra