Nest algebra

Definition

Nest algebras 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. This makes them one of the simplest examples of commutative subspace lattice algebras.

Examples

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 × 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 Sj from N then the corresponding nest algebra consists of block upper-triangular matrices.

Properties

Nest algebras are among the simplest examples of non-self-adjoint operator algebras. Furthermore, they are closed in the weak operator topology and hyperreflexive with distance constant 1.

References