Continuous operators which commute with translations, on the space of continuous functions on \(\mathbb{Z}{}_ p\). (English) Zbl 0773.47039

p-adic functional analysis, Proc. Int. Conf., Laredo/Spain 1990, Lect. Notes Pure Appl. Math. 137, 75-88 (1992).
From the author’s introduction: For a prime number \(p\), let \(Z_ p\) and \(Q_ p\) be the ring of \(p\)-adic integers and the field of \(p\)-adic numbers, respectively. Let \(K\) be a non-Archimedean nontrivially valued complete field with valuation \(|\cdot|\), and assume \(Q_ p\subseteq K\). The space of all continuous functions \(f: Z_ p\to K\), equipped with the supremum norm is \(C(Z_ p\to K)\). In this paper the linear continuous operators on the space \(C(Z_ p\to K)\), which commute with the translation operator \(E\) defined by \((Ef)(x)=f(x+1)\), are studied. If \(B_ n(x)={x\choose n}\) \((n=0,1,2,\dots)\) are binomial polynomials, it is proved that a linear continuous operator \(Q\) on \(C(Z_ p\to K)\) commutes with \(E\) if and only if the sequence \((b_ n)=(QB_ n(0))\) is bounded. A uniformly convergent expansion of an arbitrary continuous function \(f:Z_ p\to K\) is also given.
For the entire collection see [Zbl 0746.00058].


47S10 Operator theory over fields other than \(\mathbb{R}\), \(\mathbb{C}\) or the quaternions; non-Archimedean operator theory
47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)