The use of operators for the construction of normal bases for the space of continuous functions on {$V\sb q$}

Let a and q be two units of Zp, q not a root of unity, and let Vq be the closure of the setfaq n j n =0 ; 1; 2;::g. K is a non-archimedean valued eld, K contains Qp ,a ndK is complete for the valuationj:j, which extends the p-adic valuation. C(Vq ! K) is the Banach space of continuous functions from Vq to K, equipped with the supremum norm. Let E and Dq be the operators on C(Vq ! K) dened by (Ef)(x )= f(qx )a nd (Dqf)(x )=( f(qx) f(x))=(x(q 1)). We will nd all linear and continuous operators that commute withE (resp. with Dq), and we use these operators to nd normal bases for C(Vq! K).