{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "3F94J43P2C"
    },
    {
      "field": "uses",
      "target": "4TXG13F5N0"
    },
    {
      "field": "uses",
      "target": "BW678BX9ZX"
    },
    {
      "field": "uses",
      "target": "F8R88XFMQK"
    },
    {
      "field": "uses",
      "target": "QD23PJR6RP"
    },
    {
      "field": "defines",
      "target": "C37R11WREF"
    },
    {
      "field": "about",
      "target": "QD23PJR6RP"
    }
  ],
  "body": "\n## Summary\n\nA trace class operator is a bounded operator on a Hilbert space whose singular values have a finite sum.\nFor such operators the trace $\\sum_i \\ip{T e_i}{e_i}$ converges absolutely and does not depend on the orthonormal basis, so the trace of matrices extends to them.\nTrace class operators are central in the theory of Fredholm determinants.\nIn quantum mechanics, density matrices are positive trace class operators of trace $1$.\n\n## Explanations\n\n### L1 human\n\nFor a matrix, the trace is the sum of the diagonal entries.\nFor an operator on an infinite-dimensional space, this sum may diverge or depend on the chosen basis.\nTrace class operators are those small enough that the sum always converges and gives the same answer in every basis.\nFor example, the operator on square-summable sequences that multiplies the $n$-th coordinate by $1/n^2$ is trace class, with trace $\\sum 1/n^2 = \\pi^2 / 6$.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "47B10"
    ],
    "name": "Trace class operator",
    "same_as": {
      "wikidata": "Q977297",
      "wikipedia": "Trace class"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Trace_class"
      }
    ],
    "specializes": [
      "153N5DW55Z"
    ],
    "tag": "Z2CP530WDE"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/Z2CP530WDE/",
  "links": [
    {
      "field": "specializes",
      "target": "153N5DW55Z"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/Z2CP530WDE.md",
  "msc": [
    "47B10"
  ],
  "name": "Trace class operator",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/Z2C/Z2CP530WDE.md",
  "status": null,
  "tag": "Z2CP530WDE",
  "uri": "m2://Z2CP530WDE",
  "url": "https://math2.tamnd.com/wiki/trace-class-operator/",
  "writing": {
    "tag": "Z2CP530WDE",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}