Second order arithmetic means in operator ideals

TL;DR

This paper investigates conditions under which equality of second order arithmetic means implies equality of first order means in operator ideals, extending results to general ideals.

Contribution

It provides broad sufficient conditions for second order equality and inclusion cancellations in operator ideals based on growth properties of associated regularity sequences.

Findings

Second order equality does not imply first order equality in general.

Sufficient conditions are established for second order equality cancellation.

Results are extended from principal to general ideals.

Abstract

Equality of the second order arithmetic means of two principal ideals does not imply equality of their first order arithmetic means (second order equality cancellation). We provide fairly broad sufficient conditions on one of the principal ideals for this implication to hold true. We present also sufficient conditions for second order inclusion cancellations. These conditions are formulated in terms of the growth properties of the ratio of regularity sequence associated to the sequence of s-number of a generator of the principal ideal. These results are then extended to general ideals.