On sequence ideal using Orlicz function and de la Vallée Poussin mean (Q467083)

For every Orlicz function \(\phi\), there exists a complete sequence space \(\mathrm{ces}_\phi\), also called Cesàro-Orlicz sequence space, defined by \[ \mathrm{ces}_\phi = \left\{ (x_n)_{n=1}^\infty \in w: \sum_{n=1}^\infty \phi \left( \frac{1}{n} \sum_{k=1}^n \frac{|x_k|}{\sigma}\right) < \infty\text{ for some }\sigma >0\right\}, \] provided with the norm given by the infimum of the set of the numbers \(\sigma > 0\) such that the above sum is less than or equal to \(1\), where \(w\) is the set of complex (or real) sequences. From the point of view of the theory of (operator) ideals, the paper contains the definition of a new sequence ideal, the Cesàro-Orlicz sequence ideal. The concept of de la Vallée Poussin mean and Orlicz functions are the main tools used to build this ideal. The authors also prove that the Cesàro-Orlicz sequence ideal is complete with the presented norm and study its maximality and minimality.