Semi-groups of scalar type operators in Banach spaces

This paper deals with the spectral representation theorems of semi-groups of scalar type operators in Banach spaces. These results generalize the corresponding ones on semi-groups of hermitian, normal and unitary operators in Hubert spaces. In the beginning sections we study some interesting properties of a W*(\ I I |)-algebra-which generalizes the notion of an abelian von Neumann algebra to Banach spaces-and unbounded spectral operators arising out of i( )-unbounded measurable functions where E( ) is a resolution of the identity. These results are applied later to prove the spectral representation theorems on semi-groups of scalar type operators. The last theorem of this paper gives an extension of Stone's theorem on strongly continuous one parameter group of unitary operators to arbitrary Banach spaces.