THE PARSEVAL FORMULA IN THE HARMONIC ANALYSIS FOR OPERATOR AND THE SUPPORT FOR OPERATORS

【摘要】 <正> 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 (γ) and (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~*.更多还原

【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 (γ) and (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~*.更多还原