{
  "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 and [[adjoint|86B7HF79PQ]] $A^*$.\nThe *deficiency subspaces* of $A$ are the closed subspaces\n$$K_+ = \\ker(A^* - iI), \\qquad K_- = \\ker(A^* + iI)$$\nof $H$, where $\\ker$ denotes the [[kernel|0S23E6YE83]].\nThe *deficiency indices* of $A$ are the [[Hilbert dimensions|40SV748R64]] $n_+ = \\dim K_+$ and $n_- = \\dim K_-$, which are cardinal numbers, possibly infinite.\n\n## Remarks\n\nEquivalently, $K_\\pm$ is the orthogonal complement of the [[image|0WSFKKT6FF]] of $A \\pm iI$.\nSome authors exchange the names of $K_+$ and $K_-$; the pair $(n_+, n_-)$ is then reversed.\nThe operator $A$ is [[essentially self-adjoint|DT49GKR9AS]] if and only if $n_+ = n_- = 0$.\nExample: on $L^2([0,1])$, let $A = -i \\, d/dx$ on the smooth functions with compact support in $(0,1)$.\nThen $A^*$ is $-i \\, d/dx$ on the Sobolev space $H^1([0,1])$, and $A^* u = \\pm i u$ means $u' = \\mp u$.\nSo $K_+$ is spanned by $e^{-x}$ and $K_-$ by $e^{x}$, and $(n_+, n_-) = (1, 1)$.\nExample with unequal indices: on $L^2([0, \\infty))$, the same construction gives $K_+$ spanned by $e^{-x}$, while $e^{x}$ is not square integrable, so $(n_+, n_-) = (1, 0)$.\nNon-example: the deficiency indices are not defined for an operator that is not densely defined, because then the adjoint is not a well-defined operator.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "defines": "G4GMSNAX8X",
    "kind": "definition",
    "msc": [
      "47B25",
      "47A20"
    ],
    "name": "Deficiency indices",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Extensions_of_symmetric_operators"
      }
    ],
    "tag": "CSMF0WVH67",
    "uses": [
      "HM2HHE3J51",
      "H971WGMV6C",
      "86B7HF79PQ",
      "0S23E6YE83",
      "0WSFKKT6FF",
      "40SV748R64",
      "DT49GKR9AS"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/CSMF0WVH67/",
  "links": [
    {
      "field": "defines",
      "target": "G4GMSNAX8X"
    },
    {
      "field": "uses",
      "target": "HM2HHE3J51"
    },
    {
      "field": "uses",
      "target": "H971WGMV6C"
    },
    {
      "field": "uses",
      "target": "86B7HF79PQ"
    },
    {
      "field": "uses",
      "target": "0S23E6YE83"
    },
    {
      "field": "uses",
      "target": "0WSFKKT6FF"
    },
    {
      "field": "uses",
      "target": "40SV748R64"
    },
    {
      "field": "uses",
      "target": "DT49GKR9AS"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/CSMF0WVH67.md",
  "msc": [
    "47B25",
    "47A20"
  ],
  "name": "Deficiency indices",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/CSM/CSMF0WVH67.md",
  "status": null,
  "tag": "CSMF0WVH67",
  "uri": "m2://CSMF0WVH67",
  "url": "https://math2.tamnd.com/wiki/deficiency-indices-definition/",
  "writing": {
    "tag": "CSMF0WVH67",
    "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
      }
    ]
  }
}