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  "body": "\n## Statement\n\nLet $H_1$ and $H_2$ be [[Hilbert spaces|HM2HHE3J51]], let $N \\ge 1$, and let $T_1, \\dots, T_N \\colon H_1 \\to H_2$ be bounded linear operators.\nWrite $T_i^*$ for the [[adjoint|6HSJBJ8TTQ]] of $T_i$ and $\\norm{\\cdot}$ for the [[operator norm|06EKW1AKXF]].\nLet $A \\ge 0$ and assume that\n$$\\sup_{i} \\sum_{j=1}^{N} \\norm{T_i^* T_j}^{1/2} \\le A \\quad \\text{and} \\quad \\sup_{i} \\sum_{j=1}^{N} \\norm{T_i T_j^*}^{1/2} \\le A.$$\nThen\n$$\\Big\\Vert \\sum_{i=1}^{N} T_i \\Big\\Vert \\le A.$$\n\n## Remarks\n\nThe bound does not depend on $N$.\nSo let $(T_i)$ be an infinite sequence with the same two hypotheses.\nThen the partial sums are bounded by $A$, and the series $\\sum_i T_i$ converges in the strong operator topology to an operator of norm at most $A$.\n\nIdea of the proof.\nLet $T = \\sum_i T_i$ and let $m \\ge 1$.\nSince $\\norm{T}^{2m} = \\norm{(T^* T)^m}$, expand $(T^*T)^m$ as a sum of $N^{2m}$ products $T_{i_1}^* T_{i_2} T_{i_3}^* \\cdots T_{i_{2m}}$.\nBound each product in two ways, by grouping the factors in pairs $T_{i_1}^* T_{i_2}, T_{i_3}^* T_{i_4}, \\dots$ or in pairs $T_{i_2} T_{i_3}^*, \\dots$, and take the geometric mean of the two bounds.\nSumming over the indices gives $\\norm{T}^{2m} \\le N A^{2m}$, and letting $m \\to \\infty$ gives $\\norm{T} \\le A$.\n\nExample.\nIf $T_i^* T_j = 0$ and $T_i T_j^* = 0$ for all $i \\ne j$, the operators act on mutually orthogonal pieces.\nThe two hypotheses then reduce to $\\norm{T_i} \\le A$ for each $i$, and the lemma gives $\\norm{\\sum_i T_i} \\le \\max_i \\norm{T_i}$.\n\nOne hypothesis alone is not enough.\nLet $(e_k)$ be an orthonormal basis of a Hilbert space $H$ of dimension at least $N$, and let $T_i x = \\ip{x, e_1} e_i$ for $i = 1, \\dots, N$.\nThen $T_i^* T_j = 0$ for $i \\ne j$ and $\\norm{T_i^* T_i} = 1$, so the first sum equals $1$.\nBut $\\norm{T_i T_j^*} = 1$ for all $i, j$, so the second sum is $N$.\nIndeed $\\sum_i T_i$ sends $e_1$ to $e_1 + \\cdots + e_N$, so its norm is $\\sqrt{N}$.\n\n## Explanations\n\n### L1 human\n\nAdding many operators can make a large operator, even if each piece is small.\nThe lemma says that this does not happen when the pieces are almost orthogonal to each other, both on the input side and on the output side.\nThe amount of interaction between two pieces is measured by $\\norm{T_i^* T_j}$ and $\\norm{T_i T_j^*}$, and these must decay fast enough.\n\n### L2 human\n\nThe lemma is the basic tool of almost orthogonality in harmonic analysis.\nA singular integral or pseudodifferential operator is cut into pieces that live at one scale or one frequency band, and the lemma gives $L^2$ boundedness of the sum.\nA classical use is the proof of the Calderón–Vaillancourt theorem on pseudodifferential operators.\nIt only works on Hilbert spaces, since the proof uses the identity $\\norm{T}^2 = \\norm{T^*T}$.\n",
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