TOPICS

Nest and Nest Algebra

Let be a complexHilbert space, and define a nest as a set of closed subspaces of 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.

(Davidson 1988).

The nest algebra associated with the nest 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. 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.