{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "1AR19ZQQM2"
    },
    {
      "field": "uses",
      "target": "5YR4GJKPCV"
    },
    {
      "field": "uses",
      "target": "KX2XY7S3X6"
    },
    {
      "field": "defines",
      "target": "TEREJ8DA37"
    }
  ],
  "body": "\n## Summary\n\nThe resolvent of an operator $T$ is the operator-valued function $\\lambda \\mapsto (T - \\lambda I)^{-1}$, defined on the resolvent set, the complement of the spectrum.\nIt is an analytic function of $\\lambda$, and its singularities are exactly the points of the spectrum.\nThrough the resolvent, complex analysis becomes a tool of operator theory: contour integrals of the resolvent give projections and functions of $T$.\n\n## Explanations\n\n### L1 human\n\nFor a number $t$, the function $\\lambda \\mapsto 1/(t - \\lambda)$ blows up only at $\\lambda = t$.\nThe resolvent does the same for an operator: it inverts $T - \\lambda I$ wherever this is possible.\nWhere the inverse stops existing, the resolvent has a singularity, and these points form the spectrum.\nFor a diagonal matrix with entries $1$ and $2$, the resolvent is the diagonal matrix with entries $1/(1 - \\lambda)$ and $1/(2 - \\lambda)$.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "47A10"
    ],
    "name": "Resolvent",
    "same_as": {
      "wikidata": "Q1426722",
      "wikipedia": "Resolvent formalism"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Resolvent_formalism"
      }
    ],
    "tag": "J5YNNKSB9P"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/J5YNNKSB9P/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/J5YNNKSB9P.md",
  "msc": [
    "47A10"
  ],
  "name": "Resolvent",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/J5Y/J5YNNKSB9P.md",
  "status": null,
  "tag": "J5YNNKSB9P",
  "uri": "m2://J5YNNKSB9P",
  "url": "https://math2.tamnd.com/wiki/resolvent/",
  "writing": {
    "tag": "J5YNNKSB9P",
    "kind": "concept",
    "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
      }
    ]
  }
}