Geometric and probabilistic estimates for entropy and approximation numbers of operators (Q1104530)

If X and Y are Banach spaces and \(T\in L(X,Y)\), the space of bounded linear mappings of X into Y, then \(e_ n(T)\) and \(a_ n(T)\), \(n\in {\mathbb{N}}\), are the sequences of entropy and approximation numbers of T. The principal results of the paper are the following. It is proved that if X and \(Y^*\) are of type 2 then there exist constants c, d, depending only upon X and Y, such that, for any \(T\in L(X,Y)\), \[ d^{-1} e_{[nc]}(T)\leq e(T^*)\leq de_{[n/c]}(T). \] This result generalizes one of \textit{B. Carl} [Friedrich Schiller Univ. Jena 30 and 31 (1983; Zbl 0585.47015 and Zbl 0585.47016)]. An asymptotic formula is obtained for the entropy numbers of diagonal operators on a real Banach space with a 1- unconditional basis. By a combination of probabilistic averaging, which is used to obtain bounds for approximation numbers, and a method of \textit{E. D. Gluskin} [Mat. Sb., Nov. Ser. 120(162), 180-189 (1983; Zbl 0528.46015)] estimates, asymptotic in k and n, are obtained for the approximation numbers \(a_ k(Id: c^ n_ p\to c^ n_{p'})\), \((1<p<2\), \(p'=p/p-1)\), where \(c^ n_ p\) is \(L(\ell^ n_ 2,\ell^ n_ 2)\) equipped with the norm \(\| T\| =(\sum_{i}a_ ui(T)^ p)^{1/p}\), and \(a_ k(Id: \ell^ n_ p\otimes_{\pi}\ell^ n_ p\to \ell^ n_ p\otimes_{\epsilon}\ell^ n_{p'})\) (1\(\leq p\geq 2)\).