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  "body": "\n## Definition\n\nLet $H$ be a Hilbert space and let $B(H)$ be the [[algebra of bounded operators|BMDG4JQPEH]] on $H$.\nA *nest* on $H$ is a family $\\mathcal{N}$ of closed linear subspaces of $H$ such that:\n\n1. $\\set{0} \\in \\mathcal{N}$ and $H \\in \\mathcal{N}$;\n2. $\\mathcal{N}$ is a [[chain|0S8FE7ZMFM]] under inclusion: for $M, N \\in \\mathcal{N}$, $M \\subseteq N$ or $N \\subseteq M$;\n3. $\\mathcal{N}$ is complete: for every subfamily, the intersection and the closed linear span of its members belong to $\\mathcal{N}$.\n\nThe *nest algebra* of $\\mathcal{N}$ is\n$$\\operatorname{Alg} \\mathcal{N} = \\set{T \\in B(H) : T(N) \\subseteq N \\text{ for all } N \\in \\mathcal{N}},$$\nthe set of operators for which every member of $\\mathcal{N}$ is an [[invariant subspace|0ACX5QT8TE]].\n\n## Remarks\n\nExample: 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 $1 \\le k \\le n$.\nA 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.\nNon-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.\nIndeed $\\C e_1$ and $\\C e_2$ are not comparable.\nThe operators leaving this family invariant are the diagonal matrices.\nThe diagonal algebra is not a nest algebra at all, since the nests on $\\C^2$ give only $M_2(\\C)$ and algebras of dimension $3$.\nEvery nest algebra is weakly closed, contains the identity, and is reflexive.\n",
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