Abstract: In this paper a harmonic analysis for operators on a homogeneous Banach space on the additional group of real numbers is discussed. Main resu lts are as following:
1. The Parseval formula for integrable operators with respect right translation holds under some conditions; i.e., the Fourier transform for the product of opreator T and T^* at zero is equal to the integral for the product of Fourier transform T(γ) and T(γ)^* in the strong operator topelogy.
2. The finite support and finite cosupport for any beunded linear operator are both unique except the having Haar measure zero.
3. The following are equivalent:(1) The measurable set M is the supporting set for operator T; (2). The measursble set M is the cosupporting set for T^*.