On the Tameness of Power Series Space Pairs

TL;DR

This paper characterizes the tameness of power series space pairs through quasi-diagonal operators, completing previous classifications and showing that continuous tame operators between such spaces have bases.

Contribution

It provides a complete characterization of tameness for power series space pairs and links this property to the structure of operators between these spaces.

Findings

Tameness of space pairs is determined by quasi-diagonal operators.

Complete classification of power series space pairs is provided.

Continuous tame operators have bases in their ranges.

Abstract

In this paper, it is shown that the tameness of the K\"othe space pair $(\lambda^p(A),\lambda^q(B))$ is determined solely by the tameness of the family of quasi-diagonal operators defined between the pair of spaces. We use this tool to fill the gaps in characterization of pairs of power series spaces, adding to the previously established results of Dubinsky, Vogt, Nyberg and etc., and summarize this complete characterization in Table 1. As a result, we also show that the range of every continuous tame operator defined between power series spaces of infinite type has a basis.