{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $H$ be a complex [[Hilbert space|HM2HHE3J51]] and let $T \\colon H \\to H$ be a bounded linear operator with [[adjoint|6HSJBJ8TTQ]] $T^*$.\nThe operator $T$ is *hyponormal* when $T^*T - TT^*$ is a [[positive operator|77020CY5ZW]], that is,\n$$\\ip{(T^*T - TT^*)x}{x} \\ge 0 \\quad \\text{for all } x \\in H.$$\n\n## Remarks\n\nExample: for the unilateral shift $S(x_0, x_1, \\ldots) = (0, x_0, x_1, \\ldots)$ on $\\ell^2(\\N)$, one has $S^*S = I$ and $SS^*$ is the projection onto the vectors with $x_0 = 0$.\nSo $S^*S - SS^*$ is the projection onto the first coordinate, which is positive; $S$ is hyponormal.\n\nNon-example: the backward shift $S^*$ gives $(S^*)^* S^* - S^*(S^*)^* = SS^* - S^*S$, the negative of that projection, and $\\ip{(SS^* - S^*S)e_0}{e_0} = -1 \u003c 0$ for $e_0 = (1, 0, 0, \\ldots)$.\n\nEquivalent form: $T$ is hyponormal if and only if $\\norm{T^*x} \\le \\norm{Tx}$ for all $x \\in H$, since $\\ip{(T^*T - TT^*)x}{x} = \\norm{Tx}^2 - \\norm{T^*x}^2$.\nThe operator $T$ is normal exactly when both $T$ and $T^*$ are hyponormal.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "defines": "35XKRHEC3Q",
    "kind": "definition",
    "msc": [
      "47B20"
    ],
    "name": "Hyponormal operator",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Hyponormal_operator"
      }
    ],
    "tag": "CM3KYNZZJ6",
    "uses": [
      "HM2HHE3J51",
      "6HSJBJ8TTQ",
      "77020CY5ZW"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/CM3KYNZZJ6/",
  "links": [
    {
      "field": "defines",
      "target": "35XKRHEC3Q"
    },
    {
      "field": "uses",
      "target": "HM2HHE3J51"
    },
    {
      "field": "uses",
      "target": "6HSJBJ8TTQ"
    },
    {
      "field": "uses",
      "target": "77020CY5ZW"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/CM3KYNZZJ6.md",
  "msc": [
    "47B20"
  ],
  "name": "Hyponormal operator",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/CM3/CM3KYNZZJ6.md",
  "status": null,
  "tag": "CM3KYNZZJ6",
  "uri": "m2://CM3KYNZZJ6",
  "url": "https://math2.tamnd.com/wiki/hyponormal-operator-definition/",
  "writing": {
    "tag": "CM3KYNZZJ6",
    "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
      }
    ]
  }
}