## 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].

### MSC:

 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.)