On tame operators between non-archimedean power series spaces

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.