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In functional analysis, a branch of mathem … In functional analysis, a branch of mathematics, nest algebras are a class of operator algebras that generalise the upper-triangular matrix algebras to a Hilbert space context. They were introduced by Ringrose 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 . 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 -dimensional complex vector space , and let be the standard basis. For , let be the -dimensional subspace of spanned 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.nsists of block upper-triangular matrices.
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In functional analysis, a branch of mathem … In functional analysis, a branch of mathematics, nest algebras are a class of operator algebras that generalise the upper-triangular matrix algebras to a Hilbert space context. They were introduced by Ringrose and have many interesting properties. They are non-selfadjoint algebras, are closed in the weak operator topology and are reflexive. 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.sely the set of upper-triangular matrices.
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Nest algebra
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