{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "MM07SS0PB4"
    }
  ],
  "body": "\n## Summary\n\nThe Borel functional calculus assigns an operator $f(N)$ to a normal or self-adjoint operator $N$ and a Borel function $f$ on its spectrum.\nIt is defined by integrating $f$ against the projection-valued measure given by the spectral theorem.\nIt extends the continuous functional calculus to discontinuous functions, such as indicator functions, which produce the spectral projections of $N$.\n\n## Explanations\n\n### L1 human\n\nFor a diagonal matrix, applying a function to the matrix means applying it to each diagonal entry.\nThe spectral theorem says that normal operators are diagonal in a generalized sense, with a measure in place of a list of eigenvalues.\nThe Borel functional calculus applies a function to an operator in the same way, entry by entry along the spectrum.\nFor the matrix $\\operatorname{diag}(1, 2)$ and the indicator function $f$ of the set $\\set{2}$, the result is $f(N) = \\operatorname{diag}(0, 1)$, the projection onto the eigenspace of $2$.\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "kind": "concept",
    "msc": [
      "47A60",
      "47B15"
    ],
    "name": "Borel functional calculus",
    "same_as": {
      "wikidata": "Q829645",
      "wikipedia": "Borel functional calculus"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Borel_functional_calculus"
      }
    ],
    "tag": "92433H25TC"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/92433H25TC/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/92433H25TC.md",
  "msc": [
    "47A60",
    "47B15"
  ],
  "name": "Borel functional calculus",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/924/92433H25TC.md",
  "status": null,
  "tag": "92433H25TC",
  "uri": "m2://92433H25TC",
  "url": "https://math2.tamnd.com/wiki/borel-functional-calculus/",
  "writing": {
    "tag": "92433H25TC",
    "kind": "concept",
    "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
      }
    ]
  }
}