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 Mobius 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 wn = f2(n). Analogously, we show that is bounded on l2(ℕ,1/n2α) with $${\left\| {{{\cal M}_\mu }} \right\|_{2,\omega }} = {{\zeta (\alpha )} \over {\zeta (2\alpha )}}$$ , where ωn = 1/n2α, 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.