Commutants of the Euler operator and corresponding mean-periodic functions

Abstract

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=δ M in C ¹(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, unique mean-periodic solutions of Euler differential equations are characterized.

Keywords:

• Commutant
• Riesz–Markov theorem
• Invariant hyperplane
• Convolutional algebra
• Multiplier
• Cyclic element
• Mean-periodic function