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On strong convex compactness property of spaces of nonlinear operators

Published online by Cambridge University Press:  17 April 2009

Xueli Song
Affiliation:
Department of Mathematics, Xi'an Jiaotong University, Xi'an 710049, China, e-mail: [email protected], [email protected]
Jigen Peng
Affiliation:
Department of Mathematics, Xi'an Jiaotong University, Xi'an 710049, China, e-mail: [email protected], [email protected]
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The strong convex compactness property is important for property persistence of operator semigroups under perturbations. It has been investigated in the linear setting. In this paper, we are concerned with the property in the nonlinear setting. We prove that the following spaces of (nonlinear) operators enjoy the strong convex compactness property: the space of compact operators, the space of completely continuous operators, the space of weakly compact operators, the space of conditionally weakly compact operators, the space of weakly completely continuous operators, the space of demicontinuous operators, the space of weakly continuous operators and the space of strongly continuous operators. Moreover, we prove the property persistence of operator semigroups under nonlinear perturbation.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 2006

References

[1]Banach, S., Théorie des opérations liniéarires (Chelsea, New york, 1932).Google Scholar
[2]Bátkai, A. and Piazzera, S., ‘Semigroups and linear partial differential equations with delay’, J. Math. Anal. Appl. 264 (2001), 120.CrossRefGoogle Scholar
[3]Diestel, J. and Uhl-Jr, J.J., Vector measure (American Mathematical Society, Providence, R.I., 1977).CrossRefGoogle Scholar
[4]Diestel, J., Sequences and series in Banach spaces (Springer-Verlag, New York, 1984).CrossRefGoogle Scholar
[5]Doytchinov, B.D., Hrusa, W.J. and Watson, S.J., ‘On perturbations of differentiable semigroups’, Semigroup Forum 54 (1997), 100111.CrossRefGoogle Scholar
[6]Engel, K.J. and Nagel, R., One-parameter semigroups for linear evolution equations (Springer-Verlag, New York, 2000).Google Scholar
[7]Jung, M., ‘Multiplicative perturbations in semigroup theory with the (Z)-condition’, Semigroup Forum 52 (1996), 197211.CrossRefGoogle Scholar
[8]Mátrai, T., ‘On perturbations of eventually compact semigroups preserving eventual compactness’, Semigroup Forum 69 (2004), 317340.Google Scholar
[9]Miyadera, I., Nonlinear semigroups (American Mathematical Society, Providence, R.I., 1992).CrossRefGoogle Scholar
[10]Mokhtar-Kharroubi, M., ‘On the convex compactness property for the strong operator topology and related topics’, Math. Methods Appl. Sci. 27 (2004), 687701.CrossRefGoogle Scholar
[11]Nagel, R. and Piazzera, S., ‘On the regularity properties of perturbed seimgroups’, Rend. Circ. Mat. Palermo II, Suppl. 56 (1998), 99110.Google Scholar
[12]Pazy, A., Semigroups of linear operators and applications to partial differential equations (Springer-Verlag, New York, 1983).CrossRefGoogle Scholar
[13]Peng, J.G. and Xu, Z.B., ‘A novel approach to nonlinear semigroups of Lipschitz operators’, Trans. Amer. Math. Soc. 357 (2005), 409424.CrossRefGoogle Scholar
[14]Schlüchtermann, G., ‘On weakly compact operators’, Math. Ann. 292 (1992), 263266.CrossRefGoogle Scholar
[15]Voigt, J., ‘On the convex compactness property for the strong operator topology’, Note Mat. 12 (1992), 259269.Google Scholar
[16]Weis, L.W., ‘A generalization of the Vidav-Jöirgens perturbation theorem of semigroups and its application to transport theory’, J. Math. Anal. Appl. 29 (1988), 623.CrossRefGoogle Scholar
[17]Yosida, K., Functional analysis (Springer-Verlag, Berlin, 1999).Google Scholar