{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "V2H578D4RT"
    }
  ],
  "body": "\n## Statement\n\nLet $H$ be a real or complex [[Hilbert space|HM2HHE3J51]].\nLet $T$ be a [[compact operator|M2NEP1K5AE]] on $H$ with $T = T^*$, where $T^*$ is the [[adjoint|6HSJBJ8TTQ]] of $T$.\nThen there are a finite or countable orthonormal family $(e_n)_{n \\in N}$ in $H$ and nonzero real numbers $(\\lambda_n)_{n \\in N}$ such that:\n\n1. $T e_n = \\lambda_n e_n$ for every $n \\in N$;\n2. $\\abs{\\lambda_1} \\ge \\abs{\\lambda_2} \\ge \\cdots$, and $\\lambda_n \\to 0$ if $N$ is infinite;\n3. $T x = \\sum_{n \\in N} \\lambda_n \\ip{x}{e_n} e_n$ for every $x \\in H$;\n4. $H$ is the orthogonal direct sum of the [[kernel|0S23E6YE83]] $\\ker T$ and the closed span of $(e_n)$.\n\n## Remarks\n\nThe proof starts from the fact that $\\norm{T} = \\sup_{\\norm{x} = 1} \\abs{\\ip{Tx}{x}}$ for self-adjoint $T$.\nCompactness makes this supremum attained, at an [[eigenvector|0N6960W0BV]] with eigenvalue $\\pm \\norm{T}$.\nThe orthogonal complement of this eigenvector is invariant under $T$, and one repeats the argument there; the eigenvalues found tend to $0$ by the [[spectral theory of compact operators|JEPDM5B5FC]].\nAdding an [[orthonormal basis|0S9GGTR0EX]] of $\\ker T$ to $(e_n)$ gives an orthonormal basis of $H$ made of eigenvectors of $T$.\nThe nonzero [[eigenvalues|0BQAA2BK3Q]] of $T$ are exactly the numbers $\\lambda_n$, and each appears as many times as the dimension of its eigenspace, which is finite.\nExample: the operator $(Tf)(x) = \\int_0^1 \\min(x, y) f(y) \\, dy$ on $L^2([0,1])$ is compact and self-adjoint, with eigenvalues $\\big((n - \\tfrac{1}{2}) \\pi\\big)^{-2}$ and eigenfunctions $\\sqrt{2} \\sin\\big((n - \\tfrac{1}{2}) \\pi x\\big)$, $n \\ge 1$.\nWithout self-adjointness the theorem fails: the Volterra operator $(Vf)(x) = \\int_0^x f(y) \\, dy$ on $L^2([0,1])$ is compact but has no eigenvalue at all.\nIn finite dimension, this is the [[spectral theorem|06J4PXRG22]] for self-adjoint linear maps.\n\n## Explanations\n\n### L1 human\n\nA symmetric matrix can be diagonalized in an orthonormal basis of eigenvectors.\nThis theorem says the same for compact self-adjoint operators on infinite-dimensional spaces.\nThe operator acts by stretching along infinitely many perpendicular axes, by factors that shrink to zero.\n\n### L2 human\n\nThe theorem is the main tool for solving linear integral equations with symmetric kernels, and for boundary value problems such as the Sturm–Liouville problem through their Green's operators.\nIt shows that the eigenfunctions of such problems form an orthonormal basis, which justifies expansions in Fourier-type series.\nApplied to $T^* T$ for an arbitrary compact operator $T$, it gives the singular values and the Schmidt decomposition of $T$.\nFor noncompact self-adjoint operators, eigenvectors may not exist, and one needs the spectral theorem in its measure form.\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "about": [
      "M2NEP1K5AE"
    ],
    "kind": "statement",
    "msc": [
      "47B07",
      "47B15",
      "47A75"
    ],
    "name": "Spectral theorem for compact self-adjoint operators",
    "role": "theorem",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Compact_operator_on_Hilbert_space"
      }
    ],
    "tag": "A1G25R99PY",
    "uses": [
      "HM2HHE3J51",
      "M2NEP1K5AE",
      "6HSJBJ8TTQ",
      "0BQAA2BK3Q",
      "0N6960W0BV",
      "0S23E6YE83",
      "0S9GGTR0EX",
      "06J4PXRG22",
      "JEPDM5B5FC"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/A1G25R99PY/",
  "links": [
    {
      "field": "about",
      "target": "M2NEP1K5AE"
    },
    {
      "field": "uses",
      "target": "HM2HHE3J51"
    },
    {
      "field": "uses",
      "target": "M2NEP1K5AE"
    },
    {
      "field": "uses",
      "target": "6HSJBJ8TTQ"
    },
    {
      "field": "uses",
      "target": "0BQAA2BK3Q"
    },
    {
      "field": "uses",
      "target": "0N6960W0BV"
    },
    {
      "field": "uses",
      "target": "0S23E6YE83"
    },
    {
      "field": "uses",
      "target": "0S9GGTR0EX"
    },
    {
      "field": "uses",
      "target": "06J4PXRG22"
    },
    {
      "field": "uses",
      "target": "JEPDM5B5FC"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/A1G25R99PY.md",
  "msc": [
    "47B07",
    "47B15",
    "47A75"
  ],
  "name": "Spectral theorem for compact self-adjoint operators",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/A1G/A1G25R99PY.md",
  "status": null,
  "tag": "A1G25R99PY",
  "uri": "m2://A1G25R99PY",
  "url": "https://math2.tamnd.com/wiki/spectral-theorem-for-compact-self-adjoint-operators/",
  "writing": {
    "tag": "A1G25R99PY",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}