{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $H$ be a complex [[Hilbert space|HM2HHE3J51]], and let $A$ be a [[symmetric operator|H971WGMV6C]] on $H$ with dense domain $D(A)$.\nThen $A$ is [[closable|3TWMBFQJJ8]], with [[closure|TRC892PJNN]] $\\overline{A}$.\nThe operator $A$ is *essentially self-adjoint* if $\\overline{A}$ is self-adjoint, that is, $\\overline{A} = (\\overline{A})^*$, where ${}^*$ denotes the [[adjoint|86B7HF79PQ]].\n\n## Remarks\n\nThe following conditions are equivalent for a symmetric operator $A$:\n\n1. $A$ is essentially self-adjoint;\n2. $A$ has exactly one self-adjoint extension;\n3. $A^*$ is self-adjoint, and then $A^* = \\overline{A}$;\n4. the [[kernels|0S23E6YE83]] $\\ker(A^* - iI)$ and $\\ker(A^* + iI)$ are both $\\set{0}$.\n\nExample: on $L^2(\\R)$, let $A = -i \\, d/dx$ on the smooth functions with compact support.\nThen $A^*$ is $-i \\, d/dx$ on the Sobolev space $H^1(\\R)$.\nThe equation $A^* u = \\pm i u$ means $u' = \\mp u$, with solutions $c e^{\\mp x}$, which are not in $L^2(\\R)$ unless $c = 0$; so condition 4 holds.\nNon-example: on $L^2([0,1])$, let $B = -i \\, d/dx$ on the smooth functions with compact support in $(0,1)$.\nNow $e^{-x}$ and $e^{x}$ lie in $L^2([0,1])$ and solve $B^* u = i u$ and $B^* u = -i u$, so condition 4 fails, and $B$ has many self-adjoint extensions, one for each boundary condition $u(1) = e^{i \\theta} u(0)$.\nA self-adjoint operator is essentially self-adjoint; a core of a self-adjoint operator $T$ is a domain on which the restriction of $T$ is essentially self-adjoint.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "defines": "DT49GKR9AS",
    "kind": "definition",
    "msc": [
      "47B25"
    ],
    "name": "Essentially self-adjoint operator",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Self-adjoint_operator"
      }
    ],
    "tag": "Z8CW6Z2K7N",
    "uses": [
      "HM2HHE3J51",
      "H971WGMV6C",
      "3TWMBFQJJ8",
      "TRC892PJNN",
      "86B7HF79PQ",
      "0S23E6YE83"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/Z8CW6Z2K7N/",
  "links": [
    {
      "field": "defines",
      "target": "DT49GKR9AS"
    },
    {
      "field": "uses",
      "target": "HM2HHE3J51"
    },
    {
      "field": "uses",
      "target": "H971WGMV6C"
    },
    {
      "field": "uses",
      "target": "3TWMBFQJJ8"
    },
    {
      "field": "uses",
      "target": "TRC892PJNN"
    },
    {
      "field": "uses",
      "target": "86B7HF79PQ"
    },
    {
      "field": "uses",
      "target": "0S23E6YE83"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/Z8CW6Z2K7N.md",
  "msc": [
    "47B25"
  ],
  "name": "Essentially self-adjoint operator",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/Z8C/Z8CW6Z2K7N.md",
  "status": null,
  "tag": "Z8CW6Z2K7N",
  "uri": "m2://Z8CW6Z2K7N",
  "url": "https://math2.tamnd.com/wiki/essentially-self-adjoint-operator-definition/",
  "writing": {
    "tag": "Z8CW6Z2K7N",
    "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
      }
    ]
  }
}