# One-parameter semi-group

A family of operators $ T ( t) $,
$ t > 0 $,
acting in a Banach or topological vector space $ X $,
with the property

$$ T ( t + \tau ) x = \ T ( t) [ T ( \tau ) x],\ \ t, \tau > 0,\ \ x \in X. $$

If the operators $ T ( t) $ are linear, bounded and are acting in a Banach space $ X $, then the measurability of all the functions $ T ( t) x $, $ x \in X $, implies their continuity. The function $ \| T ( t) \| $ increases no faster than exponentially at infinity. The classification of one-parameter semi-groups is based on their behaviour as $ t \rightarrow 0 $. In the simplest case $ T ( t) $ is strongly convergent to the identity operator as $ t \rightarrow 0 $( see Semi-group of operators).

An important characteristic of a one-parameter semi-group is the generating operator of a semi-group. The basic problem in the theory of one-parameter semi-groups is the establishment of relations between properties of semi-groups and their generating operators. One-parameter semi-groups of continuous linear operators in locally convex spaces have been studied rather completely.

One-parameter semi-groups of non-linear operators in Banach spaces have been investigated in the case when the operators $ T ( t) $ are contractive. There are deep connections here with the theory of dissipative operators.

#### References

[1] | K. Yosida, "Functional analysis" , Springer (1980) pp. Chapt. 8, §1 MR0617913 Zbl 0435.46002 |

[2] | S.G. Krein, "Linear differential equations in Banach space" , Transl. Math. Monogr. , 29 , Amer. Math. Soc. (1971) (Translated from Russian) MR0342804 Zbl 0179.20701 |

[3] | E. Hille, R.S. Phillips, "Functional analysis and semi-groups" , Amer. Math. Soc. (1957) MR0089373 Zbl 0392.46001 Zbl 0033.06501 |

[4] | P. Butzer, H. Berens, "Semigroups of operators and approximation" , Springer (1967) MR230022 |

[5] | V. Barbu, "Nonlinear semigroups and differential equations in Banach spaces" , Ed. Academici (1976) (Translated from Rumanian) MR0390843 Zbl 0328.47035 |

[6] | E.B. Davies, "One-parameter semigroups" , Acad. Press (1980) MR0591851 Zbl 0457.47030 |

[7] | J.A. Goldstein, "Semigroups of linear operators and applications" , Oxford Univ. Press (1985) MR0790497 Zbl 0592.47034 |

#### Comments

#### References

[a1] | A. Pazy, "Semigroups of linear operators and applications to partial differential equations" , Springer (1983) MR0710486 Zbl 0516.47023 |

[a2] | Ph. Clément, H.J.A.M. Heijmans, S. Angenent, C.J. van Duijn, B. de Pagter, "One-parameter semigroups" , CWI Monographs , 5 , North-Holland (1987) MR0915552 Zbl 0636.47051 |

[a3] | H. Brezis, "Operateurs maximaux monotone et semigroups de contractions dans les espaces de Hilbert" , North-Holland (1973) MR348562 |

[a4] | J. van Casteren, "Generators of strongly continuous semigroups" , Pitman (1985) Zbl 0576.47023 |

[a5] | R. Nagel (ed.) , One-parameter semigroups of positive operators , Springer (1986) MR0839450 Zbl 0585.47030 |

**How to Cite This Entry:**

One-parameter semi-group.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=One-parameter_semi-group&oldid=48041