{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $X$ and $Y$ be [[normed spaces|03J5RAKB30]] over $\\R$ or $\\C$, and let $T \\colon X \\to Y$ be a [[continuous linear map|03TM135G0N]].\nThe operator $T$ is *invertible* if there is a continuous linear map $S \\colon Y \\to X$ such that\n$$S T = \\id_X \\quad \\text{and} \\quad T S = \\id_Y.$$\nThe map $S$ is then unique; it is the *inverse* of $T$, written $T^{-1}$.\n\n## Remarks\n\nSo $T$ is an [[inverse element|0ZNZH0EXFX]] for composition, with the inverse required to be bounded.\n\nExample: on $\\ell^2$, the diagonal operator $D(x_n) = ((1 + 1/n) x_n)$ is invertible.\nIts inverse is $D^{-1}(x_n) = (\\tfrac{n}{n+1} x_n)$, which is bounded with norm at most $1$, and both compositions are the identity.\n\nNon-example: the diagonal operator $E(x_n) = (x_n / n)$ on $\\ell^2$ is injective, but it is not invertible.\nIt is not surjective, since $(1/n)$ lies in $\\ell^2$ but $(1)$ does not, so $(1/n)$ is not in the image of $E$.\n\nIf $X$ and $Y$ are [[Banach spaces|06GZ956657]], every bijective bounded operator is invertible, by the [[bounded inverse theorem|MR5FJXR348]].\nFor an operator $T$ on one space $X$, invertibility means invertibility in the algebra of bounded operators on $X$.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "defines": "CPAD5W1CGW",
    "kind": "definition",
    "msc": [
      "47A05"
    ],
    "name": "Invertible operator",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Inverse_element"
      }
    ],
    "tag": "Y0V7AP2E09",
    "uses": [
      "03J5RAKB30",
      "03TM135G0N",
      "0ZNZH0EXFX",
      "06GZ956657",
      "MR5FJXR348"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/Y0V7AP2E09/",
  "links": [
    {
      "field": "defines",
      "target": "CPAD5W1CGW"
    },
    {
      "field": "uses",
      "target": "03J5RAKB30"
    },
    {
      "field": "uses",
      "target": "03TM135G0N"
    },
    {
      "field": "uses",
      "target": "0ZNZH0EXFX"
    },
    {
      "field": "uses",
      "target": "06GZ956657"
    },
    {
      "field": "uses",
      "target": "MR5FJXR348"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/Y0V7AP2E09.md",
  "msc": [
    "47A05"
  ],
  "name": "Invertible operator",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/Y0V/Y0V7AP2E09.md",
  "status": null,
  "tag": "Y0V7AP2E09",
  "uri": "m2://Y0V7AP2E09",
  "url": "https://math2.tamnd.com/wiki/invertible-operator-definition/",
  "writing": {
    "tag": "Y0V7AP2E09",
    "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
      }
    ]
  }
}