Some properties of Fourier transform for operators on homogeneous Banach spaces

The Fourier transform of linear operator on a general homogeneous Banach space B in L^{1}(G) for locally compact abelian group G is defined and characterized. It is proved that the Fourier transform of a linear operator is an operator valued continuous function on \hat{G} , th dual group of G, and vanishing at infinity. Convolution of function and operator is studied. Some linear operator on B is characterized as an integration of its Fourier transform over \hat{G} .