Resultant force

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

$\{\left(\begin{array}{cc}a& b\\ 0& a\end{array}\right):a,b\in \mathbb{ℂ}\}.$

This algebra is smaller than the Nest algebra

$\{\left(\begin{array}{cc}a& b\\ 0& c\end{array}\right):a,b,c\in \mathbb{ℂ}\}$

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

$\{\left(\begin{array}{ccc}a& b& 0\\ 0& a& 0\\ 0& 0& a\end{array}\right):a,b\in \mathbb{ℂ}\}$

which is equal to the set of all polynomials in

$T=\left(\begin{array}{ccc}0& 1& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right)$

and the identity is reflexive.

Hyper-reflexivity

Let $\mathcal{𝒜}$ 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

$\beta (T,\mathcal{𝒜})=\sup \{\Vert P^{\perp }TP\Vert :P\text{ is a projection and }{P}^{\perp }\mathcal{𝒜}P=(0)\}$.

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

The algebra $\mathcal{𝒜}$ is reflexive if and only if for every T in B(H):

$\beta (T,\mathcal{𝒜})=0\text{ implies that }T\text{ is in }\mathcal{𝒜}$.

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

$\beta (T,\mathcal{𝒜})\le \text{dist}(T,\mathcal{𝒜})$.

Here $\text{dist}(T,\mathcal{𝒜})$ 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 $\mathcal{𝒜}$ hyperreflexive if there is a constant K such that for every operator T in B(H),

$\text{dist}(T,\mathcal{𝒜})\le K\beta (T,\mathcal{𝒜})$.

The smallest such K is called the distance constant for $\mathcal{𝒜}$. 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

References

• William Arveson, Ten lectures on operator algebras, ISBN 0-8218-0705-6
• H. Radjavi and P. Rosenthal, Invariant Subspaces, ISBN 0-486-42822-2