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  "body": "\n## Statement\n\nLet $X$ be an infinite-dimensional complex [[Banach space|06GZ956657]] and let $K \\colon X \\to X$ be a nonzero [[compact operator|M2NEP1K5AE]].\n\n1. There is a closed linear subspace $M$ of $X$ with $M \\ne \\set{0}$ and $M \\ne X$ such that $A(M) \\subseteq M$ for every bounded linear operator $A \\colon X \\to X$ with $AK = KA$.\n2. In particular, every bounded linear operator $T \\colon X \\to X$ with $TK = KT$ has a nontrivial closed [[invariant subspace|0ACX5QT8TE]].\n\n## Remarks\n\nProof idea: let $\\mathcal{A}$ be the algebra of operators that commute with $K$, and suppose that it has no common nontrivial closed invariant subspace.\nThen the orbit $\\set{Ay : A \\in \\mathcal{A}}$ of every $y \\ne 0$ is dense.\nA compactness argument with finitely many $A_i \\in \\mathcal{A}$ and a partition of unity builds a continuous map of a closed ball into itself.\nThe [[Schauder fixed-point theorem|DCPFVVZZ54]] gives a fixed point.\nThe fixed point is an eigenvector, with eigenvalue $1$, of a compact operator $L$ that commutes with $K$.\nSo $K$ has an eigenvalue $\\lambda$ on the finite-dimensional space $\\ker(L - I)$.\nThen $\\ker(K - \\lambda I)$ is a common invariant subspace, a contradiction.\n\nThe infinite dimension is needed in part 1.\nOn $\\C^n$ with $n \\ge 2$, the identity $K = I$ is compact and commutes with every operator, and the operators on $\\C^n$ have no common nontrivial invariant subspace.\n\nExample: the Volterra operator $(Vf)(x) = \\int_0^x f(t)\\,dt$ on $L^2[0,1]$ is compact, so every operator that commutes with $V$ has a nontrivial closed invariant subspace.\n\n## Explanations\n\n### L1 human\n\nCompact operators are the operators that are closest to matrices.\nThe theorem says that a nonzero compact operator always shares a proper closed invariant subspace with every operator that commutes with it.\nSo any operator that commutes with a nonzero compact operator escapes the [[invariant subspace problem|5RF055GK4N]].\n\n### L2 human\n\nBefore this theorem, the invariant subspace problem was known for compact operators and for polynomially compact operators, with long proofs.\nLomonosov's short fixed-point argument covers all of these at once, and in arbitrary complex Banach spaces.\nIt does not settle the general problem: there are operators on Hilbert space that commute with no nonzero compact operator.\n",
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