### 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.