The use of operators for the construction of normal bases for the space of continuous functions on $$V_ q$$.(English)Zbl 0814.46070

Summary: Let $$a$$ and $$q$$ be two units of $$\mathbb{Z}_ p$$, $$q$$ not a root of unity, and let $$V_ q$$ be the closure of the set $$\{aq^ n\mid n= 0,1,2,\dots\}$$. $$K$$ is a non-Archimedean valued field, $$K$$ contains $$\mathbb{Q}_ p$$, and $$K$$ is complete for the valuation $$| .|$$, which extends the $$p$$-adic valuation. $$C(V_ q\to K)$$ is the Banach space of continuous functions from $$V_ q$$ to $$K$$, equipped with the supremum norm.
Let $$\mathcal E$$ and $$D_ q$$ be the operators on $$C(V_ q\to K)$$ defined by $$({\mathcal E}f)(x)= f(qx)$$ and $$(D_ q f)(x)= (f(qx)- f(x))/(x(q- 1))$$. We will find all linear and continuous operators that commute with $$\mathcal E$$ (resp. with $$D_ q$$), and we use these operators to find normal bases for $$C(V_ q\to K)$$.

MSC:

 46S10 Functional analysis over fields other than $$\mathbb{R}$$ or $$\mathbb{C}$$ or the quaternions; non-Archimedean functional analysis 46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces 47S10 Operator theory over fields other than $$\mathbb{R}$$, $$\mathbb{C}$$ or the quaternions; non-Archimedean operator theory
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