Abstract
A quasicomplementM to a subspaceN of a Banach spaceX is called strict ifM does not contain an infinite-dimensional subspaceM 1 such that the linear manifoldN+M 1 is closed. It is proved that ifX is separable, thenN always admits a strict quasicomplement. We study the properties of the restrictions of the operators of dense imbedding to infinite-dimensional closed subspaces of a space where these operators are defined.
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References
V. I. Gurarii and M. I. Kadets, “On minimal systems and quasicomplements in Banach spaces,”Dokl. Akad. Nauk SSSR,145, No. 2, 256–258 (1962).
J. Dixmier, “Etude sur les varietes et les operateurs de Julia,”Bull. Soc. Math. France,77, 11–101 (1949).
I. Singer,Bases in Banach Spaces. II, Springer, Berlin, etc. (1981).
V. D. Mil'man, “Geometric theory of Banach spaces. I,”Usp. Mat. Nauk,25, Issue 3, 111–170 (1970).
V. I. Gurarii, “On the slopes and openings of subspaces of a Banach space,”Teor. Funkts., Funkts. Anal. Prilozh., Issue 1, 194–204 (1965).
J. Lindenstrauss and L. Tzafriri,Classical Banach Spaces. I, Springer, Berlin, etc. (1977).
A. Pietsch,Operator Ideals, VEB Deutscher Verlag der Wissenschaften, Berlin (1978).
V. V. Shevchik, “On noncompact operators of dense imbedding in Banach spaces,”Izv. Vyssh. Uchebn. Zaved.,12, 79–81 (1988).
V. V. Shevchik, “On subspaces in a pair of Banach spaces,”Mat. Zametki,38, 545–553 (1985).
V. V. Shevchik, “Action of an imbedding operator on minimal sequences in a Banach space,”Dokl. Akad. Nauk SSSR,294, No. 5, 1072–1076 (1987).
Additional information
Zaporozhye University, Zaporozhye. Translated from Ukrainskii Matematicheskii Zhurmal, Vol. 46, No. 6, pp. 789–792, June, 1994
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Shevchik, V.V. Strict quasicomplements and the operators of dense imbedding. Ukr Math J 46, 863–867 (1994). https://doi.org/10.1007/BF02658190
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DOI: https://doi.org/10.1007/BF02658190