{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $X$ be a complex Banach space with identity map $I$, and let $T \\colon X \\to X$ be a bounded operator with [[spectrum|BMKATJ802A]] $\\sigma(T)$ and [[point spectrum|7FNVA0Q6YQ]] $\\sigma_p(T)$.\nFor $\\lambda \\in \\sigma(T) \\setminus \\sigma_p(T)$, the operator $T - \\lambda I$ is injective.\n\n1. The *continuous spectrum* $\\sigma_c(T)$ is the set of $\\lambda \\in \\sigma(T) \\setminus \\sigma_p(T)$ such that the [[image|0WSFKKT6FF]] of $T - \\lambda I$ is [[dense|0BSJSX7ME2]] in $X$.\n2. The *residual spectrum* $\\sigma_r(T)$ is the set of $\\lambda \\in \\sigma(T) \\setminus \\sigma_p(T)$ such that the image of $T - \\lambda I$ is not dense in $X$.\n\n## Remarks\n\nBy construction, $\\sigma(T)$ is the disjoint union $\\sigma_p(T) \\cup \\sigma_c(T) \\cup \\sigma_r(T)$.\nFor $\\lambda \\in \\sigma_c(T)$, the image of $T - \\lambda I$ is dense but not all of $X$, since a bounded bijection between Banach spaces has a bounded inverse.\n\nExample: for $D(x_n) = (x_n / n)$ on $\\ell^2$, $0 \\in \\sigma_c(D)$.\nIndeed $D$ is injective, it is not surjective, and its image contains every finitely supported sequence, a dense set.\n\nNon-example: for the right shift $S(x_1, x_2, \\ldots) = (0, x_1, x_2, \\ldots)$ on $\\ell^2$, $0$ is not in $\\sigma_c(S)$.\nThe image of $S$ is the closed subspace $\\set{x : x_1 = 0}$, which is not dense, so $0 \\in \\sigma_r(S)$.\n\nFor the right shift, $\\sigma_p(S) = \\emptyset$, $\\sigma_r(S)$ is the open unit disc and $\\sigma_c(S)$ is the unit circle.\nA self-adjoint operator on a Hilbert space has empty residual spectrum.\n",
  "commit": "efe85cee8dbc57d85bedbacfb97b99d1b7bee843",
  "front": {
    "defines": "YVH149HCAG",
    "kind": "definition",
    "msc": [
      "47A10"
    ],
    "name": "Continuous and residual spectrum",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Decomposition_of_spectrum_(functional_analysis)"
      }
    ],
    "tag": "Y95W7EME5Q",
    "uses": [
      "BMKATJ802A",
      "7FNVA0Q6YQ",
      "0BSJSX7ME2",
      "0WSFKKT6FF"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/Y95W7EME5Q/",
  "links": [
    {
      "field": "defines",
      "target": "YVH149HCAG"
    },
    {
      "field": "uses",
      "target": "BMKATJ802A"
    },
    {
      "field": "uses",
      "target": "7FNVA0Q6YQ"
    },
    {
      "field": "uses",
      "target": "0BSJSX7ME2"
    },
    {
      "field": "uses",
      "target": "0WSFKKT6FF"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/Y95W7EME5Q.md",
  "msc": [
    "47A10"
  ],
  "name": "Continuous and residual spectrum",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/Y95/Y95W7EME5Q.md",
  "status": null,
  "tag": "Y95W7EME5Q",
  "uri": "m2://Y95W7EME5Q",
  "url": "https://math2.tamnd.com/wiki/continuous-and-residual-spectrum-definition/",
  "writing": {
    "tag": "Y95W7EME5Q",
    "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
      }
    ]
  }
}