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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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