{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $H$ be a complex [[Hilbert space|HM2HHE3J51]] and let $S \\colon H \\to H$ be a bounded linear operator.\nThe operator $S$ is *subnormal* when there are a Hilbert space $K \\supseteq H$, with $H$ closed in $K$, and a bounded normal operator $N \\colon K \\to K$ such that\n\n1. $N(H) \\subseteq H$, that is, $H$ is an [[invariant subspace|0ACX5QT8TE]] of $N$;\n2. $Nx = Sx$ for all $x \\in H$.\n\nSuch an $N$ is a *normal extension* of $S$.\n\n## Remarks\n\nExample: the unilateral shift $S$ on $\\ell^2(\\N)$ is subnormal.\nView $\\ell^2(\\N)$ as the sequences in $\\ell^2(\\Z)$ that vanish at negative indices, and let $N$ be the bilateral shift $(Nx)_n = x_{n-1}$, which is unitary, hence normal.\n$N$ maps sequences vanishing at negative indices to sequences of the same kind, and it agrees with $S$ there.\n\nNon-example: $S = \\begin{pmatrix} 0 \u0026 1 \\\\ 0 \u0026 0 \\end{pmatrix}$ on $\\C^2$ is not subnormal.\nA subnormal operator is hyponormal, that is $S^*S - SS^* \\ge 0$, but here $S^*S - SS^* = \\mathrm{diag}(-1, 1)$, which is not positive.\n\nA normal extension $N$ can be chosen minimal: $K$ is the smallest closed subspace containing $H$ and invariant under $N$ and $N^*$.\nThe minimal normal extension is unique up to unitary equivalence.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "defines": "81MZ023K1A",
    "kind": "definition",
    "msc": [
      "47B20"
    ],
    "name": "Subnormal operator",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Subnormal_operator"
      }
    ],
    "tag": "273NZSYWZA",
    "uses": [
      "HM2HHE3J51",
      "0ACX5QT8TE"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/273NZSYWZA/",
  "links": [
    {
      "field": "defines",
      "target": "81MZ023K1A"
    },
    {
      "field": "uses",
      "target": "HM2HHE3J51"
    },
    {
      "field": "uses",
      "target": "0ACX5QT8TE"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/273NZSYWZA.md",
  "msc": [
    "47B20"
  ],
  "name": "Subnormal operator",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/273/273NZSYWZA.md",
  "status": null,
  "tag": "273NZSYWZA",
  "uri": "m2://273NZSYWZA",
  "url": "https://math2.tamnd.com/wiki/subnormal-operator-definition/",
  "writing": {
    "tag": "273NZSYWZA",
    "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
      }
    ]
  }
}