Fractional powers of momentum of a spectral distribution

Jazar, M.

doi:10.1090/S0002-9939-1995-1242090-0

Fractional powers of momentum of a spectral distribution
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Proc. Amer. Math. Soc. 123 (1995), 1805-1813 Request permission

Abstract:

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.

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Functional analysis