---
tag: N528CFDN46
kind: definition
name: Nest algebra
msc: [47L35, 47A15]
defines: ZMY9H2XENE
uses: [HM2HHE3J51, BMDG4JQPEH, 0ACX5QT8TE, 0S8FE7ZMFM]
sources:
  - url: https://en.wikipedia.org/wiki/Nest_algebra
---

## Definition

Let $H$ be a Hilbert space and let $B(H)$ be the [[algebra of bounded operators|BMDG4JQPEH]] on $H$.
A *nest* on $H$ is a family $\mathcal{N}$ of closed linear subspaces of $H$ such that:

1. $\set{0} \in \mathcal{N}$ and $H \in \mathcal{N}$;
2. $\mathcal{N}$ is a [[chain|0S8FE7ZMFM]] under inclusion: for $M, N \in \mathcal{N}$, $M \subseteq N$ or $N \subseteq M$;
3. $\mathcal{N}$ is complete: for every subfamily, the intersection and the closed linear span of its members belong to $\mathcal{N}$.

The *nest algebra* of $\mathcal{N}$ is
$$\operatorname{Alg} \mathcal{N} = \set{T \in B(H) : T(N) \subseteq N \text{ for all } N \in \mathcal{N}},$$
the set of operators for which every member of $\mathcal{N}$ is an [[invariant subspace|0ACX5QT8TE]].

## 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.