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  "body": "\n## Statement\n\nLet $X$ be a real [[Banach space|06GZ956657]] and let $K \\subseteq X$ be a closed convex cone with $K \\cap (-K) = \\set{0}$.\nAssume that $K$ is total, that is, the set $K - K = \\set{u - v : u, v \\in K}$ is dense in $X$.\nLet $T \\colon X \\to X$ be a [[compact operator|M2NEP1K5AE]] with $T(K) \\subseteq K$ and with [[spectral radius|9V7YQA85PC]] $r(T) \u003e 0$.\nThen $r(T)$ is an eigenvalue of $T$ with an [[eigenvector|0N6960W0BV]] in $K$: there is $u \\in K$, $u \\ne 0$, with $Tu = r(T) u$.\n\n## Remarks\n\nProof idea: for $\\lambda \u003e r(T)$ the resolvent $(\\lambda I - T)^{-1} = \\sum_{n \\ge 0} T^n / \\lambda^{n+1}$ maps $K$ into $K$.\nUsing totality of the cone one shows that $r(T)$ lies in the spectrum and that the resolvent blows up along vectors of $K$ as $\\lambda$ decreases to $r(T)$.\nNormalizing $(\\lambda I - T)^{-1} x$ for suitable $x \\in K$ and using compactness of $T$ to pass to a limit gives a nonzero eigenvector in $K$.\n\nExample: let $X = \\R^n$, let $K$ be the vectors with nonnegative entries, and let $T$ be a matrix with nonnegative entries and $r(T) \u003e 0$.\nThe theorem gives the weak form of the [[Perron–Frobenius theorem|62NS7PV1M9]]: $r(T)$ is an eigenvalue with a nonnegative eigenvector.\n\nThe hypothesis $r(T) \u003e 0$ is needed: the Volterra operator $(Vf)(x) = \\int_0^x f(t)\\,dt$ on $C[0, 1]$ is compact and maps nonnegative functions to nonnegative functions, but $r(V) = 0$ and $V$ has no eigenvalue at all.\n\nA stronger form holds when $K$ has nonempty interior and $T$ maps $K \\setminus \\set{0}$ into the interior of $K$.\nThen $r(T) \u003e 0$, it is a simple eigenvalue with an eigenvector in the interior of $K$, and every other eigenvalue $\\lambda$ satisfies $\\abs{\\lambda} \u003c r(T)$.\n\n## Explanations\n\n### L1 human\n\nA positive operator here is one that sends nonnegative things to nonnegative things, such as a matrix with nonnegative entries acting on vectors with nonnegative entries.\nThe theorem says that, under a compactness condition, the largest growth rate of such an operator is achieved by a nonnegative eigenvector.\nIt extends the Perron–Frobenius theorem from matrices to infinite-dimensional spaces.\n\n### L2 human\n\nThe theorem is a main tool for principal eigenvalues of elliptic operators.\nThe inverse of such an operator, with suitable boundary conditions, is compact and positive on a space of continuous functions, so it has a positive eigenfunction.\nThis gives the existence of a positive first eigenfunction of the Dirichlet Laplacian and similar results in population dynamics and bifurcation theory.\n",
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