{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $H$ be a [[Hilbert space|HM2HHE3J51]] over $\\R$ or $\\C$ and let $A \\colon H \\to H$ be a bounded linear operator.\nThe operator $A$ is *positive* when\n\n1. $A$ is self-adjoint, that is, $A^* = A$, where $A^*$ is the [[adjoint|6HSJBJ8TTQ]] of $A$;\n2. $\\ip{Ax}{x} \\ge 0$ for all $x \\in H$.\n\nOne writes $A \\ge 0$, and $A \\le B$ for self-adjoint $A$ and $B$ when $B - A \\ge 0$.\n\n## Remarks\n\nExample: for every bounded operator $T$ on $H$, the operator $A = T^*T$ is positive.\nIt is self-adjoint since $(T^*T)^* = T^*T$, and $\\ip{T^*Tx}{x} = \\ip{Tx}{Tx} = \\norm{Tx}^2 \\ge 0$.\n\nNon-example: the diagonal operator $A = \\mathrm{diag}(1, -1)$ on $\\C^2$ is self-adjoint, but $\\ip{A e_2}{e_2} = -1 \u003c 0$ for $e_2 = (0, 1)$, so condition 2 fails.\n\nOver $\\C$, condition 2 alone implies condition 1, because a bounded operator with $\\ip{Ax}{x}$ real for all $x$ is self-adjoint by polarization.\nOver $\\R$, condition 1 is needed: a rotation by $90$ degrees of $\\R^2$ has $\\ip{Ax}{x} = 0$ for all $x$ but is not self-adjoint.\nEquivalent forms: $A$ is positive if and only if $A$ is self-adjoint with spectrum in $[0, \\infty)$, and if and only if $A = B^*B$ for some bounded operator $B$.\nSome authors call these operators positive semidefinite or nonnegative.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "defines": "77020CY5ZW",
    "kind": "definition",
    "msc": [
      "47B65",
      "47B15"
    ],
    "name": "Positive operator",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Positive_operator"
      }
    ],
    "tag": "P93WWQ6BBB",
    "uses": [
      "HM2HHE3J51",
      "6HSJBJ8TTQ"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/P93WWQ6BBB/",
  "links": [
    {
      "field": "defines",
      "target": "77020CY5ZW"
    },
    {
      "field": "uses",
      "target": "HM2HHE3J51"
    },
    {
      "field": "uses",
      "target": "6HSJBJ8TTQ"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/P93WWQ6BBB.md",
  "msc": [
    "47B65",
    "47B15"
  ],
  "name": "Positive operator",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/P93/P93WWQ6BBB.md",
  "status": null,
  "tag": "P93WWQ6BBB",
  "uri": "m2://P93WWQ6BBB",
  "url": "https://math2.tamnd.com/wiki/positive-operator-definition/",
  "writing": {
    "tag": "P93WWQ6BBB",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}