Nest algebra
Encyclopedia
In functional analysis
, a branch of mathematics, nest algebras are a class of operator algebra
s that 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 operator
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, 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
-dimension
al complex
vector space
, and let 
be the standard basis
. For
, let 
be the 
-dimensional subspace
of
span
ned by the first
basis vectors 
. Let
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
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.
Functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
, a branch of mathematics, nest algebras are a class of operator algebra
Operator algebra
In functional analysis, an operator algebra is an algebra of continuous linear operators on a topological vector space with the multiplication given by the composition of mappings...
s that generalise the upper-triangular matrix algebras to a Hilbert space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
context. They were introduced by John Ringrose in the mid-1960s and have many interesting properties. They are non-selfadjoint algebras, are closed
Closed
Closed may refer to:Math* Closure * Closed manifold* Closed orbits* Closed set* Closed differential form* Closed map, a function that is closed.Other* Cloister, a closed walkway* Closed-circuit television...
in the weak operator topology
Weak operator topology
In functional analysis, the weak operator topology, often abbreviated WOT, is the weakest topology on the set of bounded operators on a Hilbert space H, such that the functional sending an operator T to the complex number is continuous for any vectors x and y in the Hilbert space.Equivalently, a...
and are reflexive
Reflexive operator algebra
In functional analysis, a reflexive operator algebra A is an operator algebra that has enough invariant subspaces to characterize it. Formally, A is reflexive if it is equal to the algebra of bounded operators which leave invariant each subspace left invariant by every operator in A.This should not...
.
Nest algebras are among the simplest examples of commutative subspace lattice algebras. Indeed, they are formally defined as the algebra of bounded operator
Bounded operator
In functional analysis, a branch of mathematics, a bounded linear operator is a linear transformation L between normed vector spaces X and Y for which the ratio of the norm of L to that of v is bounded by the same number, over all non-zero vectors v in X...
s leaving invariant
Invariant (mathematics)
In mathematics, an invariant is a property of a class of mathematical objects that remains unchanged when transformations of a certain type are applied to the objects. The particular class of objects and type of transformations are usually indicated by the context in which the term is used...
each subspace
Subspace
-In mathematics:* Euclidean subspace, in linear algebra, a set of vectors in n-dimensional Euclidean space that is closed under addition and scalar multiplication...
contained in a subspace nest, that is, a set of subspaces which is totally ordered
Total order
In set theory, a total order, linear order, simple order, or ordering is a binary relation on some set X. The relation is transitive, antisymmetric, and total...
by inclusion
Subset
In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
and is also a complete lattice
Complete lattice
In mathematics, a complete lattice is a partially ordered set in which all subsets have both a supremum and an infimum . Complete lattices appear in many applications in mathematics and computer science...
. 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
-dimensionDimension
In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...
al complex
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
, and let be the standard basisStandard basis
In mathematics, the standard basis for a Euclidean space consists of one unit vector pointing in the direction of each axis of the Cartesian coordinate system...
. For
, let be the -dimensional subspaceSubspace
-In mathematics:* Euclidean subspace, in linear algebra, a set of vectors in n-dimensional Euclidean space that is closed under addition and scalar multiplication...
of
spanLinear span
In the mathematical subfield of linear algebra, the linear span of a set of vectors in a vector space is the intersection of all subspaces containing that set...
ned by the first
basis vectors . Letthen 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
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.