Abstract
We determine the smooth points of certain spaces of bounded operatorsL(X,Y), including the cases whereX andY arel p-orc 0-direct sums of finite dimensional Banach spaces or subspaces of the latter enjoying the metric compact approximation property. We also remark that the operators not attaining their norm are nowhere dense inL(X,Y) wheneverK(X,Y) is anM-ideal inL(X,Y).
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References
D. Amir and J. Lindenstrauss. The structure of weakly compact sets in Banach spaces.Ann. of Math., 88:35–46, 1968.
E. Behrends. Sur lesM-idéaux des espaces d'opérateurs compacts. In B. Beauzamy, B. Maurey, and G. Pisier, editors,Séminaire d'Analyse Fonctionnelle 1984/85. Université Paris VI/VII, pages 11–18, 1985.
J. Bourgain. On dentability and the Bishop-Phelps property.Israel J. Math., 28:265–271, 1977.
M. Fabian and G. Godefroy. The dual of every Asplund space admits a projectional resolution of the identity.Studia Math., 91:141–151, 1988.
P. Harmand and Å. Lima. Banach spaces which areM-ideals in their biduals.Trans. Amer. Math. Soc., 283: 253–264, 1983.
P. Harmand, D. Werner, and W. Werner.M-ideals in Banach spaces and Banach algebras. In preparation.
R. B. Holmes and B. R. Kripke. Best approximation by compact operators.Indiana Univ. Math. J., 21:255–263, 1971.
R. B. Holmes, B. Scranton, and J. Ward. Approximation from the space of compact operators and otherM-ideals.Duke Math. J., 42:259–269, 1975.
F. Kittaneh and R. Younis. Smooth points of certain operator spaces.Int. Equ. and Oper. Th., 13:849–855, 1990.
Å. Lima and G. Olsen. Extreme points in duals of complex operator spaces.Proc. Amer. Math. Soc., 94:437–440, 1985.
J. Lindenstrauss. On operators which attain their norm.Israel J. Math., 1:139–148, 1963.
V. D. Milman. The geometric theory of Banach spaces I. The theory of basic and minimal systems.Russian Math. Surveys, 25.3:111–170, 1970.
E. Oja and D. Werner. Remarks onM-ideals of compact operators onX ⨁ p X. Math. Nachr., 1991. (to appear).
J. R. Partington. Norm attaining operators.Israel J. Math., 43:273–276, 1982.
R. Payá and W. Werner. An approximation property related toM-ideals of compact operators.Proc. Amer. Math. Soc., 1991. to appear.
W. Ruess and Ch. Stegall. Extreme points in duals of operator spaces.Math. Ann., 261:535–546, 1982.
K. Saatkamp. Best approximation in the space of bounded operators and its applications.Math. Ann., 250:35–54, 1980.
W. Schachermayer. Norm attaining operators and renormings of Banach spaces.Israel J. Math., 44:201–212, 1983.
Ch. Stegall. Optimization and differentiation in Banach spaces.Lin. Alg. and Appl., 84:191–211, 1986.
W. Zizler. On some extremal problems in Banach spaces.Math. Scand., 32:214–224, 1973.
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Werner, W. Smooth points in some spaces of bounded operators. Integr equ oper theory 15, 496–502 (1992). https://doi.org/10.1007/BF01200332
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DOI: https://doi.org/10.1007/BF01200332