On the Tameness of Power Series Space Pairs
Buket Can Bahadır
TL;DR
The paper shows that the tameness of a pair of power-series Köthe spaces is controlled by the tameness of a family of quasi-diagonal operators between them, enabling a complete characterization of tameness for all pairs of power-series spaces and consolidating prior results. By reducing tameness questions to quasi-diagonal operators, it derives precise criteria for pairs like $\big(\Lambda_0(\alpha),\Lambda_0(\beta)\big)$, $\big(\Lambda_0(\alpha),\Lambda_\infty(\beta)\big)$, $\big(\Lambda_\infty(\alpha),\Lambda_0(\beta)\big)$, and $\big(\Lambda_\infty(\alpha),\Lambda_\infty(\beta)\big)$, with stability of the defining sequences and finite limit points playing key roles. A major consequence is that the range of every continuous tame operator between infinite-type power-series spaces possesses a basis, and an absolute basis when the codomain is nuclear, established via a reduction to a tame operator on a suitably chosen auxiliary space and invoking Vogt’s results. These findings unify and extend the tameness landscape for power-series spaces and illuminate structural properties of their operators. The results are organized into a comprehensive Table 1 and reinforced by constructive proofs linking tameness to basis formation in ranges.
Abstract
In this paper, it is shown that the tameness of the Köthe space pair $(λ^p(A),λ^q(B))$ 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.