The notion of hermitian
operators in Hilbert space has been extended to Banach spaces by Lumer and Vidav.
Recently, Berkson has shown that a scalar type operator S in a Banach space X
can be decomposed into S = R + iJ where (i) R and J commute and (ii)
RmJn(m,n = 0,1,2,⋯) are hermitian in some equivalent norm on X. The converse is
also valid if the Banach space is reflexive. Thus we see that the scalar type operators
in a Banach space play a role analogous to the normal operators in a Hilbert
space.
In this paper, the well-known Hilbert space notion of unitary operators is suitably
extended to operators in Banach spaces and a polar decomposition is obtained for a
scalar type operator. It is further shown that this polar decomposition is unique and
characterises scalar type operators in reflexive Banach spaces. Finally, an
extension of stone’s theorem on one-parameter group of unitary operators in
Hilbert spaces is obtained (under suitable conditions) for reflexive Banach
spaces.
