{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $H$ be a complex [[Hilbert space|HM2HHE3J51]].\nLet $T \\colon D(T) \\to H$ be a [[densely defined|BCJHNHKQFG]] linear operator, where the domain $D(T)$ is a dense linear subspace of $H$.\nThe operator $T$ is *symmetric* when\n$$\\ip{Tx}{y} = \\ip{x}{Ty} \\quad \\text{for all } x, y \\in D(T).$$\n\n## Remarks\n\nExample: on $H = L^2(0, 1)$ let $D(T)$ be the smooth functions with compact support in $(0, 1)$ and $Tf = -i f'$.\nFor $f, g \\in D(T)$, integration by parts with vanishing boundary terms gives $\\int_0^1 -i f' \\bar g = \\int_0^1 f \\, \\overline{-i g'}$, so $T$ is symmetric.\n\nNon-example: on the same domain, $Sf = f'$ satisfies $\\ip{Sf}{g} = -\\ip{f}{Sg}$, so $\\ip{Sf}{g} \\ne \\ip{f}{Sg}$ whenever $\\ip{f}{Sg} \\ne 0$; it is skew-symmetric, not symmetric.\n\nAn equivalent form: $T$ is symmetric if and only if its [[adjoint|86B7HF79PQ]] $T^*$ extends $T$, written $T \\subseteq T^*$.\n$T$ is self-adjoint when moreover $T^* = T$, including equality of domains.\nBy the [[Hellinger–Toeplitz theorem|ZXJ7FJ3WCE]], a symmetric operator with $D(T) = H$ is bounded, and then it is self-adjoint.\nSome authors, and Mathlib, call a linear map $T \\colon E \\to E$ on an inner product space symmetric when the identity holds on all of $E$, without a domain.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "H971WGMV6C",
    "kind": "definition",
    "msc": [
      "47B25",
      "47B02"
    ],
    "name": "Symmetric operator",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Self-adjoint_operator"
      }
    ],
    "tag": "KR4KX354JN",
    "uses": [
      "HM2HHE3J51",
      "BCJHNHKQFG"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/KR4KX354JN/",
  "links": [
    {
      "field": "defines",
      "target": "H971WGMV6C"
    },
    {
      "field": "uses",
      "target": "HM2HHE3J51"
    },
    {
      "field": "uses",
      "target": "BCJHNHKQFG"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/KR4KX354JN.md",
  "msc": [
    "47B25",
    "47B02"
  ],
  "name": "Symmetric operator",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/KR4/KR4KX354JN.md",
  "status": null,
  "tag": "KR4KX354JN",
  "uri": "m2://KR4KX354JN",
  "url": "https://tamnd.github.io/math2/wiki/symmetric-operator-definition/",
  "writing": {
    "tag": "KR4KX354JN",
    "kind": "definition",
    "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
      }
    ]
  }
}