Fractional Powers of Momentum of a Spectral Distribution
M. Jazar · Proceedings of the American Mathematical Society · 1995
In this paper we construct fractional and imaginary powers for the positive momentum B of a spectral distribution and prove the basic properties. The main result is that for any $\alpha > 0, - {B^\alpha }$ generates a bounded strongly continuous holomorphic semigroup of angle $\frac {\pi }{2}$. In particular for $\alpha = 1$, using Stone’s generalized theorem, if iB generates a k-times integrated group of type $O(|t{|^k})$ with $\sigma (B) \subset [0, + \infty [$, then -B generates a strongly continuous holomorphic semigroup of angle $\frac {\pi }{2}$. A similar corollary is given in the regularized group situation.