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Entropy numbers of r-nuclear operators in Banach spaces of type q. (English) Zbl 0574.47018
At first it is proved that if X is a Banach space and \(T\in {\mathcal L}(X,\ell_ p)\) \((1<p\leq 2)\) has the property that \(T^*\) transforms the canonical p-stable cylindrical measure in \(\ell_{p'}\) \((p'=p/p-1)\) to the Radon probability measure in \(X^*\), then \((e_ n(T))\in \ell_{p',\infty}\), where \((e_ n(T))\) is the sequence of entropy numbers of T and \(\ell_{a,b}\), \(0<a,b\leq \infty\) denotes the Lorentz sequence space. Then it is obtained, that if \(1\leq q\leq 2\), \(1\leq u\leq \infty\), \(0\leq r<\min (q,u)\), Y is a Banach space of type q and \(T\in {\mathcal L}(X,\ell_ u)\) admits the factorization \(T={\mathcal D}_{\sigma}A\), where \(A\in {\mathcal L}(X,\ell_{\infty})\), \({\mathcal D}_{\sigma}:\ell_{\infty}\to \ell_ u\) is a diagonal operator with \(\sigma =(\sigma_ i)\in \ell_{r,t}\), then \(e_ n(T)\in \ell_{s,t}\), where \(1/s=1/r+1-1/q-1/u\). At last the main result is established: if X,Y are Banach spaces such that \(X^*\) is of type q and Y is of type p, \(0<r<1\) and T:\(X\to Y\) is an r-nuclear operator, then \((e_ n(T))\in \ell_{ns}\), where \(1/s=1+1/r-1/p-1/q\).
Reviewer: V.Tarieladze

47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
46A45 Sequence spaces (including Köthe sequence spaces)
46B25 Classical Banach spaces in the general theory
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