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  "body": "\n## Definition\n\nLet $H$ and $K$ be complex [[Hilbert spaces|HM2HHE3J51]], and let $1 \\le p \u003c \\infty$.\nA [[compact operator|M2NEP1K5AE]] $T \\colon H \\to K$ with [[singular values|QSJKPKPM7H]] $s_1(T) \\ge s_2(T) \\ge \\cdots$ belongs to the *Schatten $p$-class* $S_p(H, K)$ if\n$$\\norm{T}_p = \\Big( \\sum_{n=1}^{\\infty} s_n(T)^p \\Big)^{1/p} \u003c \\infty.$$\nThe number $\\norm{T}_p$ is the Schatten $p$-norm of $T$.\n\n## Remarks\n\nExample: on $\\ell^2$, the diagonal operator $T e_n = e_n / n$ has $s_n(T) = 1/n$, so $T \\in S_p$ for every $p \u003e 1$.\nNon-example: the same operator is not in $S_1$, because $\\sum 1/n$ diverges.\n$S_1$ is the trace class, and $S_2$ is the class of Hilbert–Schmidt operators, with $\\norm{T}_2^2 = \\sum_{i} \\norm{T e_i}^2$ for any orthonormal basis $(e_i)$ of $H$.\nFor $p \\le q$, $S_p \\subseteq S_q$ and $\\norm{T}_q \\le \\norm{T}_p$.\nEach $S_p(H) = S_p(H, H)$ is a two-sided ideal of the algebra of bounded operators on $H$, and a Banach space under $\\norm{\\cdot}_p$.\nThe finite-rank operators are dense in each $S_p$, and the operator norm satisfies $\\norm{T} \\le \\norm{T}_p$.\nSome authors also allow $0 \u003c p \u003c 1$, where $\\norm{\\cdot}_p$ is only a quasi-norm.\n",
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    "name": "Schatten class",
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  "msc": [
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  "name": "Schatten class",
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