Reflexive Operator Algebra - Examples

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{pmatrix} a&b\\ 0 & a \end{pmatrix} \ :\ a,b\in\mathbb{C}\right\}.

This algebra is smaller than the Nest algebra

\left\{ \begin{pmatrix} a&b\\ 0 & c \end{pmatrix} \ :\ a,b,c\in\mathbb{C}\right\}

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{pmatrix} a & b & 0\\ 0 & a & 0\\ 0 & 0 & a \end{pmatrix} \ :\ a,b\in\mathbb{C}\right\}

which is equal to the set of all polynomials in

T=\begin{pmatrix} 0 & 1 & 0\\ 0 & 0 & 0\\ 0 & 0 & 0 \end{pmatrix}

and the identity is reflexive.

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