{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $X$ be a [[Banach space|06GZ956657]] and let $\\mathcal{L}(X)$ be the space of bounded linear operators on $X$.\nA *$C_0$-semigroup* (strongly continuous semigroup) on $X$ is a map $T \\colon [0, \\infty) \\to \\mathcal{L}(X)$ such that:\n\n1. $T(0) = I$, the identity operator;\n2. $T(t + s) = T(t) T(s)$ for all $t, s \\ge 0$;\n3. $\\lim_{t \\to 0^+} \\norm{T(t) x - x} = 0$ for every $x \\in X$.\n\nIts *generator* is the operator $A$ given by $A x = \\lim_{t \\to 0^+} \\frac{T(t) x - x}{t}$, with domain $D(A)$ the set of $x \\in X$ for which this limit exists.\n\n## Remarks\n\nConditions 1 and 2 say that $T$ is a monoid homomorphism from $([0, \\infty), +)$ into the multiplicative [[semigroup|0XMEM406JM]] $\\mathcal{L}(X)$.\n\nExample: for a bounded operator $A$, $T(t) = e^{tA}$, the [[exponential|020M04FN1H]] series, is a $C_0$-semigroup with generator $A$; here even $\\norm{T(t) - I} \\to 0$ in [[operator norm|06EKW1AKXF]].\n\nExample: on $X = L^p(\\R)$ with $1 \\le p \u003c \\infty$, the left shifts $(T(t) f)(s) = f(s + t)$ form a $C_0$-semigroup.\nCondition 3 holds because translation is continuous in $L^p$ norm for $p \u003c \\infty$, which follows by approximation with continuous functions of compact support.\nThe generator is $A f = f'$ on the Sobolev space $W^{1,p}(\\R)$.\n\nNon-example: the same shifts on $L^\\infty(\\R)$ fail condition 3.\nFor $f$ the indicator function of $[0, 1]$ and $0 \u003c t \u003c 1$, $T(t) f - f$ equals $1$ in absolute value on $[-t, 0)$, so $\\norm{T(t) f - f}_\\infty = 1$.\n\nEvery $C_0$-semigroup satisfies $\\norm{T(t)} \\le M e^{\\omega t}$ for some constants $M \\ge 1$ and $\\omega \\in \\R$, and its generator is closed and densely defined.\n\n## Explanations\n\n### L1 human\n\nA $C_0$-semigroup is a time-evolution made of linear maps that compose correctly and start continuously from the identity.\nIts generator is the derivative at time $0$, which plays the role of $A$ in $e^{tA}$.\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "defines": "EVF180QJXY",
    "kind": "definition",
    "msc": [
      "47D06",
      "34G10"
    ],
    "name": "C0-semigroup",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/C0-semigroup"
      }
    ],
    "tag": "ZJF5XVY63D",
    "uses": [
      "06GZ956657",
      "06EKW1AKXF",
      "0XMEM406JM",
      "020M04FN1H"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/ZJF5XVY63D/",
  "links": [
    {
      "field": "defines",
      "target": "EVF180QJXY"
    },
    {
      "field": "uses",
      "target": "06GZ956657"
    },
    {
      "field": "uses",
      "target": "06EKW1AKXF"
    },
    {
      "field": "uses",
      "target": "0XMEM406JM"
    },
    {
      "field": "uses",
      "target": "020M04FN1H"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/ZJF5XVY63D.md",
  "msc": [
    "47D06",
    "34G10"
  ],
  "name": "C0-semigroup",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/ZJF/ZJF5XVY63D.md",
  "status": null,
  "tag": "ZJF5XVY63D",
  "uri": "m2://ZJF5XVY63D",
  "url": "https://math2.tamnd.com/wiki/c0-semigroup-definition/",
  "writing": {
    "tag": "ZJF5XVY63D",
    "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
      }
    ]
  }
}