Entropy of \(C(X)\)-valued operators and diverse applications (Q1411815)

The entropy numbers of a subset \(M\) of a metric space \((X,d)\) are defined as \(\varepsilon _n(M) := \inf \{\varepsilon > 0 : N(M,\varepsilon)\leq n\}\), where \(N(M,\varepsilon) := \inf\{N\in \mathbb N : \exists s_1,\dots ,s_N \in X\), \(M\subset \bigcup_{k=1}^{N}B(s_k,\varepsilon)\}\) and \(B(s,\varepsilon)\) stands for the closed ball in \(X\) of center \(s\) and radius \(\varepsilon > 0.\) The dyadic entropy numbers of the set \(M\) are given by \(e_n(M) := \varepsilon_{2^{n-1}}(M)\), \(n\in \mathbb N,\) and the dyadic entropy numbers of a bounded linear operator \(T\) between two Banach spaces \(E,F\) are defined as \(e_n(T) := e_n(T(B_E))\), \(n\in \mathbb N\) [see the book \textit{B. Carl} and \textit{I. Stephani}, Entropy, compactness and the approximation of operators, Cambridge University Press (1990; Zbl 0705.47017)]. The aim of the present paper is to emphasize how the geometry of the Banach space \(E\), the entropy numbers \(\varepsilon _n(X)\) of the compact metric space \((X,d)\), and the smoothness properties of an operator \(T: E\to C(X)\), expressed in terms of its modulus of continuity \(\omega(T,\delta) = \sup\{|(Tx)(s)-(Tx)(t)|: s,t\in X\), \(d(s,t)\leq \delta\), \(x\in E\), \(\|x\|\leq 1\},\) all together shape the asymptotic behaviour of the dyadic entropy numbers of \(T.\) The study is motivated by the universality of the space \(C[a,b]\) in the sense that, for any compact linear operator \(T:E\to F\), there exists a compact linear operator \(S:E\to C[a,b]\) such that \(2^{-1} e_n(S) \leq e_n(T) \leq 2e_n(S)\), \(\forall n\in \mathbb N.\) An important class of operators to which the abstract results apply is formed by the abstract kernel operators \(T_K : E\to C(X)\), defined by \( (T_Kx)(s) = \langle x,K(s)\rangle,\) where \(K\) (the abstract kernel) is a continuous mapping from the compact metric space \(X\) to the dual \(E'\) of \(E\). It turns out that \(\|T_K\|= \|K\|\), and every compact linear operator \(T:E\to C(X)\) can be obtained this way from an appropriate abstract kernel \(K\). Other applications are given to the study of the eigenvalue distributions of operators and to the metric entropy of convex hulls of precompact sets in Banach spaces.