The Grothendieck property in Lorentz sequence spaces (Q802105)

Let d(a,p) denote the Lorentz sequence space [see \textit{P. G. Casazza} and \textit{B. L. Lin}, Bull. Inst. Math., Acad. Sinica 2, 233-240 (1974; Zbl 0297.46011)] with \(1\leq p<\infty\) and a norming sequence \(a=(a_ k)_ 1^{\infty}.\) Theorem 1. If \(1\leq p<2\), then the space d(a,p) possesses the Grothendieck property if and only if \[ \sum^{n}_{k=1}(a^ q_ k)^{1/q}/\sum^{n}_{k=1}a_ k=O(n^{-p/2}). \] Theorem 2. If \(p\geq 2\), then the dual space of d(a,p) has the Grothendieck property. (We say that a space X possesses the Grothendieck property if the space of all continuous operators from \(c_ 0\) into X coincides with the space of all 2-absolutely summing operators from \(c_ 0\) into X.)