{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "3SKRBF4KG1"
    },
    {
      "field": "defines",
      "target": "3VCD0T7QNG"
    }
  ],
  "body": "\n## Summary\n\nA Toeplitz operator on the [[Hardy space|PNNNFJVF6W]] $H^2$ multiplies a function by a bounded symbol $\\varphi$ on the circle and then projects back onto $H^2$.\nIn the basis $1, z, z^2, \\ldots$ its matrix is the infinite [[Toeplitz matrix|GKY2WSHE26]] of the Fourier coefficients of $\\varphi$, constant along diagonals.\nToeplitz operators link operator theory with complex analysis: their spectra, Fredholm indices and invertibility are read from the symbol.\n\n## Explanations\n\n### L1 human\n\nThink of $H^2$ as the functions on the circle whose Fourier series has no negative terms.\nMultiplying by a function $\\varphi$ may create negative terms, so you throw them away.\nWhat remains is the Toeplitz operator $T_\\varphi$.\nWith $\\varphi(e^{it}) = e^{it}$ nothing is thrown away and $T_\\varphi$ is the shift $f \\mapsto z f$.\nWith $\\varphi(e^{it}) = e^{-it}$ the constant term is lost and one gets the backward shift.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "47B35",
      "30H10"
    ],
    "name": "Toeplitz operator",
    "same_as": {
      "wikidata": "Q7812935",
      "wikipedia": "Toeplitz operator"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Toeplitz_operator"
      }
    ],
    "tag": "GQ0X3R3WM2"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/GQ0X3R3WM2/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/GQ0X3R3WM2.md",
  "msc": [
    "47B35",
    "30H10"
  ],
  "name": "Toeplitz operator",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/GQ0/GQ0X3R3WM2.md",
  "status": null,
  "tag": "GQ0X3R3WM2",
  "uri": "m2://GQ0X3R3WM2",
  "url": "https://math2.tamnd.com/wiki/toeplitz-operator/",
  "writing": {
    "tag": "GQ0X3R3WM2",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}