Semigroups and scalar-type operators in Banach spaces (Q1328336)

This is a compilation from the authors' abstract and introduction: We extend the main result of \textit{A. R. Sourour} [Trans. Am. Math. Soc. 200, 207-232 (1974; Zbl 0304.47034), Theorem 5.3] on the representation of a \(C_ 0\)-semigroup of scalar-type operators (in Dunford's sense) in a weakly complete Banach space (under identical conditions) to the case of an arbitrary Banach space. Sourour's method of polar factorization of the semigroup into two factors of semigroups, one with nonnegative spectra and the other with spectra in the unit circle, applies here too and reduces the problem to the representation of a strongly continuous group of ``circled'' scalar-type operators, which is obtained in Theorem 1. The main result (Theorem 2) is that if \(\{T_ t: t\geq 0\}\) is a \(C_ 0\)-semigroup of scalar-type operators and if the resolutions of the identity \(E_ t\) for \(T_ t\) are uniformly bounded in norm, then the generator is of scalar-type with resolution of the identity \(E\) for which \(T_ t= \int \exp(t\lambda) E(d\lambda)\) for \(t\geq 0\). In the case when either \(\{T_ t\}\) is periodic or the Banach space is of special type (such that sums and products of scalar-type operators are also scalar- type) and the spectra \(\sigma(T_ t)\) are all contained in the unit circle, this sufficient condition can be relaxed significantly.