Nest and Nest Algebra

Let H be a complex Hilbert space, and define a nest as a set N of closed subspaces of H satisfying the conditions:

1. 0,H in N,

2. If N_1,N_2 in N, then either N_1 subset= N_2 or N_2 subset= N_1,

3. If {N_i}_(i in I) subset= N,then intersection _(i in I)T_i in N,

4. If {N_i}_(i in I) subset= N ,then the norm closure of the linear span of union _(i in I)N_i lies in N.

(Davidson 1988).

The nest algebra associated with the nest N is the set T(N)={T in B(H):T(N) subset= N for all N in N}.

For example, consider an orthonormal basis {e_j:j=1,2,...} of a separable Hilbert space H. Put N_k=span{e_1,...,e_k}. Then N={N_k:k=1,2,...} union {0,H} is a nest and the associated nest algebra T(N) is the algebra of operators whose matrix representation with respect to {e_j} is upper triangular.

This entry contributed by Mohammad Sal Moslehian

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Moslehian, Mohammad Sal. "Nest and Nest Algebra." From MathWorld--A Wolfram Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/NestandNestAlgebra.html

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