Boundedness and Spectrum of Multiplicative Convolution Operators Induced by Arithmetic Functions

In this paper, we consider a multiplicative convolution operator Mf acting on a Hilbert spaces l^2(N,ω;). In particular, we focus on the operators M1 and Mμ, where μ, is the Mobius function. We investigate conditions on the weight ω under which the operators M1 and Mμ are bounded. We show that for a positive and completely multiplicative function f,M1 is bounded on l^2(N, f^2) if and only if ||f||1 <∞, in which case ||M1||2,ω=||f||1, where ωn = f^2(n). Analogously, we show that Mμ is bounded on l^2(N, 1/n^2α) with ||M1||2,ω=ζ(α)/ζ(2α),where ωn= 1 /n^2α,α> 1. As an application, we obtain some results on the spectrum of M1^*M1 and M^*μMμ. Moreover, von Neumann algebra generated by a certain family of bounded operators is also considered.

In this paper, we consider a multiplicative convolution operator Mf acting on a Hilbert spaces ■2（N, ω）. In particular, we focus on the operators M1 and Mμ, where μ is the M?bius function.We investigate conditions on the weight ω under which the operators M1 and Mμ are bounded. We show that for a positive and completely multiplicative function f, M1 is bounded on ■2（N, f2） if and only if ‖f‖1 < ∞, in which case ‖M1‖2,ω = ‖f‖1, where ωn = f2（n）. Analogously, we show that Mμ is bounded on ■2(N, 1/n2α) with ‖Mμ‖2,ω =ζ（α）/ζ（2α）, where ωn = 1/n2α, α > 1. As an application,we obtain some results on the spectrum of M1* M1 and Mμ*Mμ. Moreover, von Neumann algebra generated by a certain family of bounded operators is also considered.