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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 $$1/s=1+1/r-1/p-1/q$$.
 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