Chemistry - Nest algebra

In functional analysis, nest algebras are a class of operator algebras which generalise the upper-triangular matrix algebras to a Hilbert space context. They were introduced by John Ringrose in the mid-1960s and have many interesting properties. They are non-selfadjoint algebras, are closed in the weak operator topology and are reflexive.

Nest algebras are among the simplest examples of commutative subspace lattice algebras . Indeed, they are formally defined as the algebra of bounded operators leaving invariant each subspace contained in a subspace nest, that is, a set of subspaces which is totally ordered by inclusion. Since the orthogonal projections corresponding to the subspaces in a nest commute, nests are commutative subspace lattices.

By way of an example, let us apply this definition to recover the finite-dimensional upper-triangular matrices. Let us work in the $n$ -dimensional complex vector space $\mathbb{C}^n$ , and let $e_1,e_2,\dots,e_n$ be the standard basis. For $j=0,1,2,\dots,n$ , let $S j$ be the $j$ -dimensional subspace of $\mathbb{C}^n$ spanned by the first $j$ basis vectors $e_1,\dots,e_j$ . Let

$N=\{ (0)=S_0, S_1, S_2, \dots, S_{n-1}, S_n=\mathbb{C}^n \}$ ;

then $N$ is a subspace nest, and the corresponding nest algebra of $n\times n$ complex matrices $M$ leaving each subspace in $N$ invariant -- that is, satisfying $MS\subseteq S$ for each $S$ in $N$ -- is precisely the set of upper-triangular matrices.

If we omit one or more of the subspaces S_j from N then the corresponding nest algebra consists of block upper-triangular matrices.

Properties

Nest algebras are hyperreflexive with distance constant 1.