A novel dual approach to nonlinear semigroups of Lipschitz operators

Authors:
Jigen Peng and Zongben Xu

Journal:
Trans. Amer. Math. Soc. **357** (2005), 409-424

MSC (2000):
Primary 47H20; Secondary 47D06

Published electronically:
August 11, 2004

MathSciNet review:
2098102

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Lipschitzian semigroup refers to a one-parameter semigroup of Lipschitz operators that is strongly continuous in the parameter. It contains -semigroup, nonlinear semigroup of contractions and uniformly -Lipschitzian semigroup as special cases. In this paper, through developing a series of Lipschitz dual notions, we establish an analysis approach to Lipschitzian semigroup. It is mainly proved that a (nonlinear) Lipschitzian semigroup can be isometrically embedded into a certain -semigroup. As application results, two representation formulas of Lipschitzian semigroup are established, and many asymptotic properties of -semigroup are generalized to Lipschitzian semigroup.

**1.**Richard F. Arens and James Eells Jr.,*On embedding uniform and topological spaces*, Pacific J. Math.**6**(1956), 397–403. MR**0081458****2.**Viorel Barbu,*Nonlinear semigroups and differential equations in Banach spaces*, Editura Academiei Republicii Socialiste România, Bucharest; Noordhoff International Publishing, Leiden, 1976. Translated from the Romanian. MR**0390843****3.**Ola Bratteli and Derek W. Robinson,*Operator algebras and quantum statistical mechanics. Vol. 1*, Springer-Verlag, New York-Heidelberg, 1979. 𝐶*- and 𝑊*-algebras, algebras, symmetry groups, decomposition of states; Texts and Monographs in Physics. MR**545651****4.**J. Czipszer and L. Gehér,*Extension of functions satisfying a Lipschitz condition*, Acta Math. Acad. Sci. Hungar.**6**(1955), 213–220 (English, with Russian summary). MR**0071493****5.**Edward Brian Davies,*One-parameter semigroups*, London Mathematical Society Monographs, vol. 15, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1980. MR**591851****6.**J. R. Dorroh and J. W. Neuberger,*A theory of strongly continuous semigroups in terms of Lie generators*, J. Funct. Anal.**136**(1996), no. 1, 114–126. MR**1375155**, 10.1006/jfan.1996.0023**7.**David J. Downing and William O. Ray,*Uniformly Lipschitzian semigroups in Hilbert space*, Canad. Math. Bull.**25**(1982), no. 2, 210–214. MR**663616**, 10.4153/CMB-1982-029-5**8.**Einar Hille and Ralph S. Phillips,*Functional analysis and semi-groups*, American Mathematical Society Colloquium Publications, vol. 31, American Mathematical Society, Providence, R. I., 1957. rev. ed. MR**0089373****9.**Ronald Larsen,*Functional analysis: an introduction*, Marcel Dekker, Inc., New York, 1973. Pure and Applied Mathematics, No. 15. MR**0461069****10.**Isao Miyadera,*Nonlinear semigroups*, Translations of Mathematical Monographs, vol. 109, American Mathematical Society, Providence, RI, 1992. Translated from the 1977 Japanese original by Choong Yun Cho. MR**1192132****11.**A. Pazy,*Semigroups of linear operators and applications to partial differential equations*, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. MR**710486****12.**Ji Gen Peng and Zong Ben Xu,*The Lipschitz dual space of a Banach space and its applications*, Acta Math. Sinica (Chin. Ser.)**42**(1999), no. 1, 61–70 (Chinese, with English and Chinese summaries). MR**1695122****13.**Jigen Peng and Si-Kit Chung,*Laplace transforms and generators of semigroups of operators*, Proc. Amer. Math. Soc.**126**(1998), no. 8, 2407–2416. MR**1469432**, 10.1090/S0002-9939-98-04603-6**14.**Q. Zheng, Strongly Continuous Semigroups of Linear Operators (in Chinese). The Press of Huazhong University of Science and Technology, Wuhan, China, 1994.

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
47H20,
47D06

Retrieve articles in all journals with MSC (2000): 47H20, 47D06

Additional Information

**Jigen Peng**

Affiliation:
Research Center for Applied Mathematics, Xi’an Jiaotong University, Xi’an 710049, People’s Republic of China

Email:
jgpeng@mail.xjtu.edu.cn

**Zongben Xu**

Affiliation:
Research Center for Applied Mathematics, Xi’an Jiaotong University, Xi’an 710049, People’s Republic of China

Email:
zbxu@mail.xjtu.edu.cn

DOI:
https://doi.org/10.1090/S0002-9947-04-03635-9

Keywords:
Lipschitz operator,
Lipschitzian semigroup,
generator,
Lipschitz dual semigroup,
$C^{*}_{0}$-semigroup

Received by editor(s):
October 8, 2003

Published electronically:
August 11, 2004

Additional Notes:
This work was supported by the Natural Science Foundation of China under contract no. 10101019

Article copyright:
© Copyright 2004
American Mathematical Society