{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $X$ be a [[Banach space|06GZ956657]] and let $T$ be a linear [[operator|KRJX5RA0PW]] in $X$ with domain $D(T)$.\nA linear operator $V$ in $X$ is *relatively bounded with respect to $T$*, or *$T$-bounded*, if:\n\n1. $D(T) \\subseteq D(V)$, and\n2. there are constants $a, b \\ge 0$ with $\\norm{Vx} \\le a \\norm{x} + b \\norm{Tx}$ for all $x \\in D(T)$.\n\nThe infimum of the numbers $b$ for which some $a$ makes clause 2 hold is the *$T$-bound* of $V$.\nThe operator $T + V$ with domain $D(T)$ is a *perturbation* of $T$.\nPerturbation theory of linear operators studies which properties of $T$ pass to $T + V$, or to $T + \\kappa V$ for small $\\kappa$.\n\n## Remarks\n\nExample: every bounded operator $V$ on $X$ is $T$-bounded with $T$-bound $0$, since $\\norm{Vx} \\le \\norm{V}\\norm{x}$.\nNon-example: on $L^2(\\R)$, let $T = -i \\, d/dx$ with domain the Sobolev space $H^1(\\R)$, and let $V = -d^2/dx^2$ with domain $H^2(\\R)$.\nClause 1 fails, since $H^1(\\R)$ is not contained in $H^2(\\R)$.\n\nA basic result is the Kato–Rellich theorem.\nIf $T$ is [[self-adjoint|0NEGN5CRSC]] on a Hilbert space and $V$ is symmetric and $T$-bounded with $T$-bound less than $1$, then $T + V$ is self-adjoint on $D(T)$.\nAnother is [[Weyl's theorem|X9XRGMDCQZ]]: a compact perturbation does not change the [[essential spectrum|FRZG0E15SK]] of a self-adjoint operator.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "defines": "JTZK8DZ257",
    "kind": "definition",
    "msc": [
      "47A55",
      "47A10"
    ],
    "name": "Perturbation theory of linear operators",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Perturbation_theory"
      }
    ],
    "tag": "9HNTG5C2XX",
    "uses": [
      "06GZ956657",
      "KRJX5RA0PW"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/9HNTG5C2XX/",
  "links": [
    {
      "field": "defines",
      "target": "JTZK8DZ257"
    },
    {
      "field": "uses",
      "target": "06GZ956657"
    },
    {
      "field": "uses",
      "target": "KRJX5RA0PW"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/9HNTG5C2XX.md",
  "msc": [
    "47A55",
    "47A10"
  ],
  "name": "Perturbation theory of linear operators",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/9HN/9HNTG5C2XX.md",
  "status": null,
  "tag": "9HNTG5C2XX",
  "uri": "m2://9HNTG5C2XX",
  "url": "https://math2.tamnd.com/wiki/perturbation-theory-of-linear-operators-definition/",
  "writing": {
    "tag": "9HNTG5C2XX",
    "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
      }
    ]
  }
}