A theorem of Hardy-Bennett-type
For a power \(p>0\) define the following sequence spaces. \[ \begin{aligned} \lambda(p)&:=\left\{{\mathbf x}:\sum\lambda_n \left(\sum_{k=1}^n| x_k| \right)^p<\infty\right\},\\ \Lambda(p)&:=\left\{{\mathbf x}: \sum_{k=1}^n| x_k| ^p=O(\Lambda_n^{1-p})\right\},\end{aligned} \] where \(\lambda=\{\lambda_n\}\) and \(\Lambda=\{\Lambda_n\}\) are given positive sequences such that \[ n\Lambda_n\leq K \Lambda_n \quad (n\in{\mathbb N}),\qquad\qquad n^\beta\Lambda_n\leq Lm^\beta\Lambda_m \quad (n\geq m) \] hold for some constants \(K=K(\lambda)>0\), \(\beta>0\) and \(L=L(\beta,\Lambda)>0\). The main result of the paper states that, for \(p>1\), a sequence \(\mathbf x\) belongs to \(\lambda(p)\) if and only if it admits a factorization \(\mathbf x={\mathbf y}\cdot{\mathbf z}\) with \({\mathbf y}\in\ell^p\) and \({\mathbf z}\in\Lambda(p^*)\), where \(p^*\) is the conjugate power to \(p\). In the particular case \(\lambda_n=n^{-p}\) the result so obtained reduces to that of \textit{G. Bennett} [Mem. Am. Math. Soc. 576 (1996; Zbl 0857.26009)].
scientific article; zbMATH DE number 1226461