Abstract
For C a bounded, injective operator with dense image, we define a C-regularized spectral distribution. This produces a functional calculus, f → f(B), from C∞(ℝ) into the space of closed densely defined operators, such that f(B)C is bounded when f has compact support. As an analogue of Stone's theorem, we characterize certain regularized spectral distributions as corresponding to generators of polynomially bounded C-regularized groups. We represent the regularized spectral distribution in terms of the regularized group and in terms of the C-resolvent. Applications include the Schrodinger equation with potential, and symmetric hyperbolic systems, all on Lp(ℝn) (1≤p<∞), Co(ℝn), BUC(ℝn), or any space of functions where translation is a bounded strongly continuous group.
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