Constructing Banach ideals using upper $\ell_p$-estimates

TL;DR

This paper introduces a new family of Banach operator ideals based on upper bb_p-estimates for weakly null sequences, expanding the understanding of operator classifications with specific parameters.

Contribution

It constructs novel operator ideals  _{\u03bb_p}^{(\u221e,)}  using upper bb_p-estimates, showing their distinctness and Banach ideal properties for certain parameters.

Findings

The new ideals contain completely continuous operators.

They are distinct for different parameter choices.

Existence of an ideal norm making  _{\u03bb_p}^{(\u221e,1)} a Banach ideal.

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 $1\leq p\leq\infty$ and $1\leq\xi\leq\omega_1$. These classes contain the completely continuous operators, and are distinct for all choices $1\leq p\leq\infty$ and, when $p\neq 1$, for all choices $\xi\neq\omega_1$. For the case $\xi=1$, there exists an ideal norm $\|\cdot\|_{(p,1)}$ on the class $\mathcal{WD}_{\ell_p}^{(\infty,1)}$ under which it forms a Banach ideal.