On tame operators between non-archimedean power series spaces

Abstract

Let p ∈ {1,∞}. We show that any continuous linear operator T from A 1(a) to A p (b) is tame, i.e., there exists a positive integer c such that sup x ‖T x ‖ k /|x| ck < ∞ for every k ∈ ℤ. Next we prove that a similar result holds for operators from A ∞(a) to A p (b) if and only if the set M b,a of all finite limit points of the double sequence (b i /a j )i,j∈ℤ is bounded. Finally we show that the range of every tame operator from A ∞(a) to A ∞(b) has a Schauder basis.