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The r-nuclearity (0 r ≦ 1) of GAUSSian covariances

✍ Scribed by Thomas Kühn; Werner Linde

Publisher
John Wiley and Sons
Year
1980
Tongue
English
Weight
458 KB
Volume
95
Category
Article
ISSN
0025-584X

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✦ Synopsis

cowariance, if there is a RADON BOREL measure p on E satisfying J e""@'dp(z) =exp ( -@a, a)/2), UEE' .

E

One of the main problem5 in the theory of GAussian measures on BANACH spaces is to characterize the class of all GAussian covariances. For instance, it is known that in spaces of type 2 an operator is a GAussian covariance, if and only if it is positive, symmetric and nuclear (cf. [7]). In the first part of this paper we investigate BANACH spaces where each positive, symmetric, r-nuclear operator (0 -=r s 1) is a GAuSSian covariance. It turns out that these are exactly the spaces of type 2r. The second part of the paper deals with the following question: Do there exist infinite dimensional BANACH spaces such that each GAussian mvariance is r-nuclear for some fixed r -= 1 ? In general we do not know the answer, but it is shown that for a large class of spaces (including the 9p-spacea, l -= p s = ) the answer is negative. In tihe case of Ci-spaces this question reduces to an old problem of GROTHENDIECK about the composition of nuclear operators.

0. Preliminaries

An operator T from a HILBERT space H into a BANACH space E is called y-RADoNifying if R = TT' is a GAussian covariance. It is known that the class of all y-RADoNifying operators from H into E is a BANACR space with respect to the norm zY, defined by where T denotes any complete orthonormal system (CONS) in H, ( y i ) denotes a sequence of independent standard GAussian random variables and the sign E stands for the mathematicR1 expectation. Some special lty-norms were estimated

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