Isomorphic embeddings and harmonic behaviour of smooth operators (Q1766476)

Conditions for embeddings \(\ell^p\) and \(c_0\) into a Banach space \(Y\) are given. Namely, if \(c_0^+\subset Y\) then \(c_0\subset Y\). Let \(1<p<\infty\). If \(\ell^p_+\subset Y\) then \(\ell^p\subset Y\). This result is applied to the study of highly smooth operators from \(\ell^p\) into~\(Y\) (\(p/2\notin {\mathbb N}\)). Let \(Y\) be a Banach space, \(1\leqslant p<\infty\), \(p/2\notin {\mathbb N}\) and let \(C=C^{[p],\alpha}(B_{\ell^p},Y)\) for some \(1\geqslant \alpha >p-[p]\) if \(p\notin {\mathbb N}\), or \(C=C^{p,+} (B_{\ell^p},Y)\) if \(p\) is an odd integer. Then either all operators in~\(C\) have a harmonic behaviour or \(\ell^{p/k}\subset Y\) for some \(1\leqslant k\leqslant [p]\).