{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "A1G25R99PY"
    }
  ],
  "body": "\n## Statement\n\nLet $X$ be a complex [[Banach space|06GZ956657]], let $T \\colon X \\to X$ be a [[compact operator|M2NEP1K5AE]], and let $\\sigma(T)$ be its [[spectrum|BMKATJ802A]].\nThen the following hold.\n\n1. Every $\\lambda \\in \\sigma(T)$ with $\\lambda \\ne 0$ is an [[eigenvalue|0BQAA2BK3Q]] of $T$, and its eigenspace $\\ker(T - \\lambda I)$ is finite-dimensional.\n2. For $\\lambda \\ne 0$, the [[image|0WSFKKT6FF]] of $T - \\lambda I$ is closed, and its codimension in $X$ equals $\\dim \\ker(T - \\lambda I)$.\n3. The set $\\sigma(T)$ is finite or countable, and $0$ is its only possible accumulation point.\n4. If $X$ is infinite-dimensional, then $0 \\in \\sigma(T)$.\n\n## Remarks\n\nThe key tool is the Riesz lemma: in an infinite-dimensional normed space, the unit ball contains a sequence of unit vectors at mutual distance at least $1/2$.\nIf $\\ker(T - \\lambda I)$ were infinite-dimensional for some $\\lambda \\ne 0$, such a sequence in it would be mapped to $\\lambda$ times itself, which has no convergent subsequence, against compactness.\nA similar argument shows that, for each $\\varepsilon \u003e 0$, only finitely many eigenvalues satisfy $\\abs{\\lambda} \\ge \\varepsilon$, which gives part 3.\nPart 2 is the Fredholm alternative: for $\\lambda \\ne 0$, the equation $Tx - \\lambda x = y$ is solvable for every $y$ as soon as the homogeneous equation has only the solution $x = 0$.\nPart 4 holds because a compact operator on an infinite-dimensional space cannot have a bounded inverse.\nExample: on $\\ell^2$, the diagonal operator $T(x_n) = (x_n / n)$ is compact, and $\\sigma(T) = \\set{1/n : n \\ge 1} \\cup \\set{0}$, where $0$ is not an eigenvalue.\nThe compactness hypothesis matters: the identity on an infinite-dimensional space has the eigenvalue $1$ with infinite-dimensional eigenspace.\n\n## Explanations\n\n### L1 human\n\nCompact operators are the infinite-dimensional operators that behave most like matrices.\nThis theorem says that their spectrum looks like that of a matrix, apart from the point $0$.\nAway from $0$, the spectrum consists of eigenvalues with finitely many independent eigenvectors each, and these eigenvalues can only pile up at $0$.\n\n### L2 human\n\nThis is the Riesz–Schauder theory, the abstract form of Fredholm's results on integral equations.\nIt applies to integral operators with continuous kernels, which are compact on spaces of continuous or square-integrable functions.\nFor compact self-adjoint operators on a Hilbert space, it combines with orthogonality to give the spectral theorem for compact self-adjoint operators.\nIn the language of Fredholm theory, part 2 says that $T - \\lambda I$ is a Fredholm operator of index $0$ for each $\\lambda \\ne 0$.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "about": [
      "M2NEP1K5AE",
      "BMKATJ802A"
    ],
    "kind": "statement",
    "msc": [
      "47B07",
      "47A10"
    ],
    "name": "Spectral theory of compact operators",
    "role": "theorem",
    "same_as": {
      "wikidata": "Q7575213",
      "wikipedia": "Spectral theory of compact operators"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Spectral_theory_of_compact_operators"
      }
    ],
    "tag": "JEPDM5B5FC",
    "uses": [
      "06GZ956657",
      "M2NEP1K5AE",
      "BMKATJ802A",
      "0BQAA2BK3Q",
      "0S23E6YE83",
      "0WSFKKT6FF"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/JEPDM5B5FC/",
  "links": [
    {
      "field": "about",
      "target": "M2NEP1K5AE"
    },
    {
      "field": "about",
      "target": "BMKATJ802A"
    },
    {
      "field": "uses",
      "target": "06GZ956657"
    },
    {
      "field": "uses",
      "target": "M2NEP1K5AE"
    },
    {
      "field": "uses",
      "target": "BMKATJ802A"
    },
    {
      "field": "uses",
      "target": "0BQAA2BK3Q"
    },
    {
      "field": "uses",
      "target": "0S23E6YE83"
    },
    {
      "field": "uses",
      "target": "0WSFKKT6FF"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/JEPDM5B5FC.md",
  "msc": [
    "47B07",
    "47A10"
  ],
  "name": "Spectral theory of compact operators",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/JEP/JEPDM5B5FC.md",
  "status": null,
  "tag": "JEPDM5B5FC",
  "uri": "m2://JEPDM5B5FC",
  "url": "https://math2.tamnd.com/wiki/spectral-theory-of-compact-operators/",
  "writing": {
    "tag": "JEPDM5B5FC",
    "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
      }
    ]
  }
}