Boundedness and Spectrum of Multiplicative Convolution Operators Induced by Arithmetic Functions

Abstract

In this paper, we consider a multiplicative convolution operator ${{\cal M}_f}$ acting on a Hilbert spaces ℓ²(ℕ,ω). In particular, we focus on the operators ${{\cal M}_1}$ and ${{\cal M}_\mu }$, where μ is the Möbius function. We investigate conditions on the weight ω under which the operators ${{\cal M}_1}$ and ${{\cal M}_\mu }$ are bounded. We show that for a positive and completely multiplicative function f, ${{\cal M}_1}$ is bounded on ℓ²(ℕ,f²)if and only if ∥f∥₁ < ∞, in which case ∥M₁ ∥₂,ω = ∥f∥₁, where w_n = f²(n). Analogously, we show that is bounded on ℓ²(ℕ,1/n^2α) with ${\left\| {{{\cal M}_\mu }} \right\|_{2,\omega }} = {{\zeta (\alpha )} \over {\zeta (2\alpha )}}$, where ω_n = 1/n^2α, α > 1. As an application, we obtain some results on the spectrum of ${\cal M}_1^ * {{\cal M}_1}$ and ${\cal M}_\mu ^ * {{\cal M}_\mu }$. Moreover, von Neumann algebra generated by a certain family of bounded operators is also considered.