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 Hilbert spaces. In the beginning sections we study
some interesting properties of a W∗(∥⋅∥)-algebra-which generalizes the notion of an
abelian von Neumann algebra to Banach spaces-and unbounded spectral
operators arising out of E(⋅)-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.
