2000 Mathematics Subject Classification: 44A35; 42A75; 47A16, 47L10, 47L80The Dunkl operators.* Supported by the Tunisian Research Foundation under 04/UR/15-02

Dimovski, Ivan

Hristov, Valentin

Sifi, Mohamed

Fractional Calculus and Applied Analysis

Bulgarian Digital Mathematics Library at IMI-BAS

Commutants of the Dunkl Operators in C(R)

The Dunkl operators Dkf(x) = df(x) dx +k f(x) − f(−x) x, k ≥ 0, are con-sidered in the space C1 = C1(R) of the smooth functions on R = (−∞,∞), and the operators M: C(R)  → C(R) with M: C1(R)  → C1(R) such that MDk = DkM in C1(R) are characterized (C(R) being the space of contin-uous functions on R). Next, for a non-zero linear functional Φ: C(R) → C the continuous linear operators M with the invariant hyperplane Φ{f}  = 0 and commuting with Dk in it are also characterized. Further, mean-periodic functions for Dk with respect to the functional Φ are introduced and it is proved that they form an ideal in a corresponding convolutional algebra (C(R), ∗). As an application the mean-periodic solutions of differential-difference equations of the form P (Dk)y = f with a polynomial P are considered

Commutants of the Dunkl Operators in C(R)

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