In functional analysis, a branch of mathematics, nest algebras are a class of operator algebras that generalise the

upper-triangular matrix In mathematics, a triangular matrix is a special kind of square matrix. A square matrix is called if all the entries ''above'' the main diagonal are zero. Similarly, a square matrix is called if all the entries ''below'' the main diagonal are ...

algebras to a Hilbert space context. They were introduced by and have many interesting properties. They are non- selfadjoint algebras, are

closed Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, ...

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 operator In functional analysis and operator theory, a bounded linear operator is a linear transformation L : X \to Y between topological vector spaces (TVSs) X and Y that maps bounded subsets of X to bounded subsets of Y. If X and Y are normed vector s ...

s 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 Commute, commutation or commutative may refer to: * Commuting, the process of travelling between a place of residence and a place of work Mathematics * Commutative property, a property of a mathematical operation whose result is insensitive to th ...

, 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 Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...

vector space

\mathbb^n

, and let

e_1,e_2,\dots,e_n

be the

standard basis In mathematics, the standard basis (also called natural basis or canonical basis) of a coordinate vector space (such as \mathbb^n or \mathbb^n) is the set of vectors whose components are all zero, except one that equals 1. For example, in the ...

. For

j=0,1,2,\dots,n

, let

S_j

be the

j

-dimensional subspace of

\mathbb^n

spanned by the first

j

basis vectors

e_1,\dots,e_j

. Let :

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 ''S_j'' from ''N'' then the corresponding nest algebra consists of block upper-triangular matrices.

Properties

* Nest algebras are hyperreflexive with distance constant 1.

References

* {{DEFAULTSORT:Nest Algebra Operator theory Operator algebras

Properties

See also

References