Nest algebra

Remarks

Example: let $H = \C^n$ with standard basis $e_1, \ldots, e_n$, and let $\mathcal{N}$ consist of $N_0 = \set{0}$ and $N_k = \operatorname{span}(e_1, \ldots, e_k)$ for $1 \le k \le n$. A matrix $T$ maps each $N_k$ into itself exactly when $T e_j \in N_j$ for all $j$, that is, when $T$ is upper triangular, so $\operatorname{Alg} \mathcal{N}$ is the algebra of upper triangular matrices. Non-example: in $\C^2$, the family $\set{0}, \C e_1, \C e_2, \C^2$ satisfies clauses 1 and 3 but not clause 2. Indeed $\C e_1$ and $\C e_2$ are not comparable. The operators leaving this family invariant are the diagonal matrices. The diagonal algebra is not a nest algebra at all, since the nests on $\C^2$ give only $M_2(\C)$ and algebras of dimension $3$. Every nest algebra is weakly closed, contains the identity, and is reflexive.