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  "body": "\n## Definition\n\nLet $a \u003c b$ and let $K \\colon [a, b] \\times [a, b] \\to \\C$ be a [[continuous kernel|ZM5G144PZG]].\nLet $T$ be the integral operator $(T x)(s) = \\int_a^b K(s, t) \\, x(t) \\, dt$ of an [[integral equation|R3SG1SN7JQ]] with kernel $K$.\nThe *trace* of $T$, or of the kernel $K$, is the number\n$$\\operatorname{tr} K = \\int_a^b K(s, s) \\, ds.$$\n\n## Remarks\n\nExample: for $K(s, t) = \\min(s, t)$ on $[0, 1]$, the kernel is continuous and $\\operatorname{tr} K = \\int_0^1 s \\, ds = 1/2$.\nNon-example: the weakly singular kernel $K(s, t) = \\abs{s - t}^{-1/2}$ on $[0, 1]$ is not continuous, and it is infinite on the diagonal, so the formula gives no number.\nFor a degenerate kernel $K(s, t) = \\sum_{i=1}^n u_i(s) v_i(t)$, the trace is $\\sum_{i=1}^n \\int_a^b u_i(s) v_i(s) \\, ds$.\nThis is the [[trace|0XC1RDBY7A]] of the $n \\times n$ matrix with entries $\\int_a^b v_i(s) u_j(s) \\, ds$, which represents $T$ on the span of $u_1, \\dots, u_n$.\nThe traces $\\int_a^b K_m(s, s) \\, ds$ of the iterated kernels $K_m$, $m \\ge 2$, appear in the power series of the Fredholm determinant.\nA continuous kernel need not define a [[trace class operator|Z2CP530WDE]] on $L^2(a, b)$; when it does, the number above equals the operator trace.\n",
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