Entropy numbers of operators factoring through general diagonal operators (Q744336)

Let \(X, Y\) be Banach spaces and \(L(X,Y)\) be the space of all bounded linear operators from \(X\) to \(Y\). For \(S\in L(X,Y)\), the \(n\)th dyadic entropy number is defined by \[ e_n(S)=\inf\{ \varepsilon\geq 0\mid \exists\; y_1, \dots, y_{2^{n-1}}\in Y:\; S(B_X(0,1)) \subseteq \cup_{i=1}^{2^{n-1}}B_Y(y_i,\varepsilon)\}, \] where \(B_Z(z,r)\) denotes the closed ball in the Banach space \(Z\) with center at \(z \in Z\) and radius \(r\geq 0\). The rate of decay of the sequence \((e_n(S))_{n\geq 1}\) can be interpreted as a degree of non-compactness of \(S\). In particular, \(S\) is compact if and only if \(\lim_{n\to \infty}e_n(S)=0\). This paper is devoted to the study of entropy numbers of those operators \(S\in L(l_u,Y)\) which admit a factorization \(S=TD_\sigma\), where \(D_\sigma:\; l_u \to \l_v\) is a diagonal operator determined by a sequence \(\sigma=(\sigma_n(S))_{n\geq 1}\) and \(T\in L(l_v,Y)\). Actually, the study is restricted to the case when the sequence \(\sigma\) belongs to some generalized Lorentz sequence space.