One-parameter semi-group

A family of operators , , acting in a Banach or topological vector space , with the property

If the operators are linear, bounded and are acting in a Banach space , then the measurability of all the functions , , implies their continuity. The function increases no faster than exponentially at infinity. The classification of one-parameter semi-groups is based on their behaviour as . In the simplest case is strongly convergent to the identity operator as (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 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
[2]	S.G. Krein, "Linear differential equations in Banach space" , Transl. Math. Monogr. , 29 , Amer. Math. Soc. (1971) (Translated from Russian)
[3]	E. Hille, R.S. Phillips, "Functional analysis and semi-groups" , Amer. Math. Soc. (1957)
[4]	P. Butzer, H. Berens, "Semigroups of operators and approximation" , Springer (1967)
[5]	V. Barbu, "Nonlinear semigroups and differential equations in Banach spaces" , Ed. Academici (1976) (Translated from Rumanian)
[6]	E.B. Davies, "One-parameter semigroups" , Acad. Press (1980)
[7]	J.A. Goldstein, "Semigroups of linear operators and applications" , Oxford Univ. Press (1985)

References

[a1]	A. Pazy, "Semigroups of linear operators and applications to partial differential equations" , Springer (1983)
[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)
[a3]	H. Brezis, "Operateurs maximaux monotone et semigroups de contractions dans les espaces de Hilbert" , North-Holland (1973)
[a4]	J. van Casteren, "Generators of strongly continuous semigroups" , Pitman (1985)
[a5]	R. Nagel (ed.) , One-parameter semigroups of positive operators , Springer (1986)