Commutants of the euler operator
Ivan H. Dimovski, Valentin Z. Hristov · 2006
Here the Euler operator δ = t d dt is considered in the space C = C(R+),R+ = (0,∞), and the operators M : C → C such that Mδ = δM in C(R+) are characterized. 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 δ in it are also characterized. Further, mean-periodic functions for δ 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 Euler differential equations are characterized.