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Difference between revisions of "One-parameter semi-group"

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====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  K. Yosida,  "Functional analysis" , Springer  (1980)  pp. Chapt. 8, §1</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S.G. Krein,  "Linear differential equations in Banach space" , ''Transl. Math. Monogr.'' , '''29''' , Amer. Math. Soc.  (1971)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  E. Hille,  R.S. Phillips,  "Functional analysis and semi-groups" , Amer. Math. Soc.  (1957)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  P. Butzer,  H. Berens,  "Semigroups of operators and approximation" , Springer  (1967)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  V. Barbu,  "Nonlinear semigroups and differential equations in Banach spaces" , Ed. Academici  (1976)  (Translated from Rumanian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  E.B. Davies,  "One-parameter semigroups" , Acad. Press  (1980)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  J.A. Goldstein,  "Semigroups of linear operators and applications" , Oxford Univ. Press  (1985)</TD></TR></table>
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<table><TR><TD valign="top">[1]</TD> <TD valign="top">  K. Yosida,  "Functional analysis" , Springer  (1980)  pp. Chapt. 8, §1 {{MR|0617913}} {{ZBL|0435.46002}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S.G. Krein,  "Linear differential equations in Banach space" , ''Transl. Math. Monogr.'' , '''29''' , Amer. Math. Soc.  (1971)  (Translated from Russian) {{MR|0342804}} {{ZBL|0179.20701}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  E. Hille,  R.S. Phillips,  "Functional analysis and semi-groups" , Amer. Math. Soc.  (1957) {{MR|0089373}} {{ZBL|0392.46001}} {{ZBL|0033.06501}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  P. Butzer,  H. Berens,  "Semigroups of operators and approximation" , Springer  (1967) {{MR|230022}} {{ZBL|}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  V. Barbu,  "Nonlinear semigroups and differential equations in Banach spaces" , Ed. Academici  (1976)  (Translated from Rumanian) {{MR|0390843}} {{ZBL|0328.47035}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  E.B. Davies,  "One-parameter semigroups" , Acad. Press  (1980) {{MR|0591851}} {{ZBL|0457.47030}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  J.A. Goldstein,  "Semigroups of linear operators and applications" , Oxford Univ. Press  (1985) {{MR|0790497}} {{ZBL|0592.47034}} </TD></TR></table>
  
  
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====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Pazy,  "Semigroups of linear operators and applications to partial differential equations" , Springer  (1983)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  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)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  H. Brezis,  "Operateurs maximaux monotone et semigroups de contractions dans les espaces de Hilbert" , North-Holland  (1973)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  J. van Casteren,  "Generators of strongly continuous semigroups" , Pitman  (1985)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  R. Nagel (ed.) , ''One-parameter semigroups of positive operators'' , Springer  (1986)</TD></TR></table>
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Pazy,  "Semigroups of linear operators and applications to partial differential equations" , Springer  (1983) {{MR|0710486}} {{ZBL|0516.47023}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  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) {{MR|0915552}} {{ZBL|0636.47051}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  H. Brezis,  "Operateurs maximaux monotone et semigroups de contractions dans les espaces de Hilbert" , North-Holland  (1973) {{MR|348562}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  J. van Casteren,  "Generators of strongly continuous semigroups" , Pitman  (1985) {{MR|}} {{ZBL|0576.47023}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  R. Nagel (ed.) , ''One-parameter semigroups of positive operators'' , Springer  (1986) {{MR|0839450}} {{ZBL|0585.47030}} </TD></TR></table>

Revision as of 12:12, 27 September 2012

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 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=28256
This article was adapted from an original article by S.G. Krein (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article