Entropy numbers of convex hulls in Banach spaces and applications

Abstract

In recent time much attention has been devoted to the study of entropy of convex hulls in Hilbert and Banach spaces and their applications in different branches of mathematics. In this paper we show how the rate of decay of the dyadic entropy numbers of a precompact set A of a Banach space X of type p, 1<p≤2, reflects the rate of decay of the dyadic entropy numbers of the absolutely convex hull aco(A) of A. Our paper is a continuation of the paper (Carl et al., 2013), where this problem has been studied in the Hilbert space case. We establish optimal estimates of the dyadic entropy numbers of aco(A) in the non-critical cases where the covering numbers N(A,ε) of A by ε-balls of X satisfy the Lorentz condition ∫0∞(log2(N(A,ε)))s/rdεs<∞ for 0<r<p′, 0<s<∞ or ∫0∞(log2(2+log2(N(A,ε))))−αs(log2(N(A,ε)))s/rdεs<∞ for p′<r<∞, 0<s≤∞ and α∈R, with the usual modifications in the case s=∞. The integral here is an improper Stieltjes integral and p′ is given by the Hölder condition 1/p+1/p′=1. It turns out that, for fixed s, the entropy of the absolutely convex hull drastically changes if the parameter r crosses the point r=p′. It is still an open problem what happens if r=p′ and 0<s<∞. However, in the case s=∞ we consider also the critical case r=p′ and, especially, the Hilbert space case r=2.We use the results for estimating entropy and Kolmogorov numbers of diverse operators acting from a Banach space whose dual space is of type p or, especially, from a Hilbert space into a C(M) space. In particular, we get entropy estimates of operators factoring through a diagonal operator and of abstract integral operators as well as of weakly singular convolution operators. Moreover, estimates of entropy and Kolmogorov numbers of the classical and generalized Riemann–Liouville operator are established, complementing and extending results in the literature.