Nest algebra

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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 and is also a complete lattice. 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.