Abstract
We study properties of bounded sets in Banach spaces, connected with the concept of equimeasurability introduced by A. Grothendieck. We introduce corresponding ideals of operators and find characterizations of them in terms of continuity of operators in certain topologies. The following result (Corollary 9) follows from the basic theorems: Let T be a continuous linear operator from a Banach space X to a Banach space Y. The following assertions are equivalent:
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1)
T is an operator of type RN;
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2)
for any Banach space Z, for any number p, p > 0, and any p-absolutely summing operator U:Z → X the operator TU is approximately p-Radonifying;
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3)
for any Banach space Z and any absolutely summing operator U:Z → X the operator TU is approximately 1-Radonifying.
We note that the implication I)⇒2), is apparently new even if the operator T is weakly compact.
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Literature cited
O. I. Reinov, Dokl. Akad. Nauk SSSR,220, No. 3, 528–531 (1975).
Seminaire Laurent Schwartz 1969/70, Centre de Math., Ecole Polytechn., Paris (1970).
L. Schwartz, J. Fac. Univ. Tokyo, Sec. 1A,18, No. 2, 139–286 (1971).
Additional information
Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 73, pp. 224–228, 1977.
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Reinov, O.I. Certain classes of sets in Banach spaces and a topological characterization of operators of type RN. J Math Sci 34, 2156–2159 (1986). https://doi.org/10.1007/BF01741593
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DOI: https://doi.org/10.1007/BF01741593