This is an announcement for the paper "Class of operators determined by ordinal indices" by Kevin Beanland, Ryan Causey, Daniel Freeman, and Ben Wallis.

Abstract: We introduce and study the Bourgain index of an operator between two Banach spaces. In particular, we study the Bourgain $\ell_p$ and $c_0$ indices of an operator. Several estimates for finite and infinite direct sums are established. We define classes determined by these indices and show that some of these classes form operator ideals. We characterize the ordinals which occur as the index of an operator and establish exactly when the defined classes are closed. We study associated indices for non-preservation of $\ell_p^\xi$ and $c_0^\xi$ spreading models and indices characterizing weak compactness of operators between separable Banach spaces. We also show that some of these classes are operator ideals and discuss closedness and distinctness of these classes.

Archive classification: math.FA

Mathematics Subject Classification: 46B28

Remarks: 45 pages

Submitted from: kbeanland@gmail.com

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/1507.06285

or

http://arXiv.org/abs/1507.06285