The algebra of smooth functions of rapid descent

Abstract

A bounded operator with the spectrum lying in a compact set V ⊂ R, has C∞(V ) functional calculus. On the other hand, an operator H acting on a Hilbert space H, admits a C(R) functional calculus if H is self-adjoint. So in a Banach space setting, we really desire a large enough intermediate topological algebra A, with C∞ 0 (R) ⊂ A ⊆ C(R) such that spectral operators or some sort of their restrictions, admit a A functional calculus. In this paper we construct such an algebra of smooth functions on R that decay like (√ 1 + x2) β as |x| → ∞, for some β < 0. Among other things, we prove that C∞ c (R) is dense in this algebra. We demonstrate that important functions like x 7→ e x are either in the algebra or can be extended to functions in the algebra. We characterize this algebra fully.