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On the relationship between \(\gamma_ p\)-Radonifying operators and other operator ideals in Banach spaces of stable type p. (English) Zbl 0668.47017
Let E be a Banach space, p a real number with \(1<p<2\), and \((\Omega,\Sigma,\mu)\) a \(\sigma\)-finite measure space. A bounded operator T from \(L_ p(\mu)\) into E is said to be \(\gamma_ p\)-Radonifying if \(\exp (-\| T'x'\|^ p)\) is the characteristic function of a Radon measure on E. The authors study the relationship between such operators and various operator ideals. For example, the condition “E is of stable type p and isomorphic to a subspace of a quotient of some \(L_ p\) space” is equivalent to the condition “an operator T is \(\gamma_ p\)- Radonifying iff its dual \(T'\) is p-integral”.
Reviewer: E.Azoff
MSC:
47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
46B20 Geometry and structure of normed linear spaces
47L10 Algebras of operators on Banach spaces and other topological linear spaces
46B25 Classical Banach spaces in the general theory
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