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  "body": "\n## Definition\n\nLet $M$ be a closed manifold of dimension $n$ with a smooth positive density, and let $E$ be a Hermitian vector bundle over $M$.\nLet $A \\colon \\Gamma(E) \\to \\Gamma(E)$ be an [[elliptic operator|MT9P2X85V8]] of order $m \u003e 0$ that is formally [[self-adjoint|0NEGN5CRSC]] and positive: $\\int_M \\ip{Au}{u} \u003e 0$ for every nonzero $u \\in \\Gamma(E)$.\nThen $L^2(E)$ has an orthonormal basis $(\\varphi_j)_{j \\ge 1}$ of smooth eigensections, $A \\varphi_j = \\lambda_j \\varphi_j$, whose [[eigenvalues|0BQAA2BK3Q]] satisfy $0 \u003c \\lambda_1 \\le \\lambda_2 \\le \\cdots$ and $\\lambda_j \\to \\infty$.\nFor $s \\in \\C$, the *complex power* $A^s$ is the operator\n$$A^s u = \\sum_{j \\ge 1} \\lambda_j^{s} \\, \\ip{u}{\\varphi_j} \\, \\varphi_j, \\qquad u \\in \\Gamma(E),$$\nwhere $\\lambda_j^s = e^{s \\log \\lambda_j}$ with the real logarithm.\n\n## Remarks\n\nThe powers satisfy $A^s A^t = A^{s+t}$, $A^0 = \\id$ and $A^1 = A$, and $A^{-1}$ is the inverse of $A$.\nSeeley proved that $A^s$ is a classical [[pseudo-differential operator|FF27N0BWTP]] of order $ms$.\nFor $\\operatorname{Re} s \u003e n/m$, the operator $A^{-s}$ is [[trace class|Z2CP530WDE]], and the spectral zeta function\n$$\\zeta_A(s) = \\operatorname{tr} A^{-s} = \\sum_{j \\ge 1} \\lambda_j^{-s}$$\nis holomorphic there.\nIt extends to a meromorphic function on $\\C$ with at most simple poles, and it is holomorphic at $s = 0$.\nThis gives the [[zeta-regularized|VJCJYXGTDY]] determinant $\\det A = e^{-\\zeta_A'(0)}$.\nMore generally, Seeley defined $A^s$ by a contour integral of $\\lambda^s (A - \\lambda)^{-1}$ for elliptic $A$ whose principal symbol has no eigenvalues on some ray from $0$, without assuming self-adjointness.\n\nExample: on the circle $S^1 = \\R / 2\\pi\\Z$ with $dx$, let $A = 1 - d^2/dx^2$, which is elliptic of order $2$ and positive.\nIts eigensections are $e^{ikx}/\\sqrt{2\\pi}$, $k \\in \\Z$, with eigenvalues $1 + k^2$, so $A^{-s} e^{ikx} = (1 + k^2)^{-s} e^{ikx}$.\nThen $\\zeta_A(s) = \\sum_{k \\in \\Z} (1 + k^2)^{-s}$ converges for $\\operatorname{Re} s \u003e \\frac{1}{2} = \\frac{n}{m}$ with $n = 1$ and $m = 2$.\n\nNon-example: the operator $-d^2/dx^2$ on $S^1$ is elliptic and self-adjoint but not positive, since it kills the constants.\nThe formula would need $0^s$, which is not defined for all $s$; one first restricts to the orthogonal complement of the kernel.\n\n## Explanations\n\n### L1 human\n\nComplex powers of a positive elliptic operator raise each eigenvalue to the power $s$, keeping the eigenfunctions, just as for a positive matrix.\n",
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