Constructing Banach ideals using upper ${\ell }_p$-estimates

Abstract

Using upper ${\ell }_p$-estimates for normalized weakly null sequence images, we describe a new family of operator ideals ${\mathcal{WD}}_{{\ell }_p}^{(\infty ,\xi )}$ with parameters $1\le p\le \infty$ and $1\le \xi \le {\omega }_1$. These classes contain the completely continuous operators, and are distinct for all choices $1\le p\le \infty$ and, when $p\ne 1$, for all choices $\xi \ne {\omega }_1$. For the case $\xi =1$, there exists an ideal norm $\Vert \cdot {\Vert }_{(p,1)}$ on the class ${\mathcal{WD}}_{{\ell }_p}^{(\infty ,1)}$ under which it forms a Banach ideal.

References in corpus (1)

• A norm for Tsirelson's Banach space