On tame operators between non-archimedean power series spaces

Let p ∈ {1,∞}. We show that any continuous linear operator T from A1(a) to Ap(b) is tame, i.e., there exists a positive integer c such that supx ‖Tx‖k/|x|ck < ∞ for every k ∈ ℤ. Next we prove that a similar result holds for operators from A∞(a) to Ap(b) if and only if the set Mb,a of all finite limit points of the double sequence (bi/aj)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.