Multipliers and Wiener-Hopf operators on weighted L ^p spaces

Abstract

We study multipliers M (bounded operators commuting with translations) on weighted spaces L ^p _ω (ℝ), and establish the existence of a symbol µ _M for M, and some spectral results for translations S _t and multipliers. We also study operators T on the weighted space L ^p _ω (ℝ⁺) commuting either with the right translations S _t , t ∈ ℝ⁺, or left translations P ⁺ S _−t , t ∈ ℝ⁺, and establish the existence of a symbol µ of T. We characterize completely the spectrum σ(S _t ) of the operator S _t proving that

$\sigma (S_t ) = \{ z \in \mathbb{C}:|z| \leqslant e^{t\alpha _0 } \} ,$

where α ₀ is the growth bound of (S _t ) _t≥0. A similar result is obtained for the spectrum of (P ⁺ S _−t ), t ≥ 0. Moreover, for an operator T commuting with S _t , t ≥ 0, we establish the inclusion

, where $\mathcal{O}$ = {z ∈ ℂ: Im z < α ₀}.