### Abstract:

A closed densely defined operator H, on a Banach space X, whose
spectrum is contained in R and satisfies
(z − H)−1
≤ c
z α
| z|β
∀ z ∈ R with z α :=
|z|2 + 1 (1)
for some α , β ≥ 0; c > 0, is said to be of (α, β) − type R (notation
introduced in [10]). For (α, α+1) − type R operators we constructed
an A-functional calculus in a more general Banach space setting (where
A is the algebra of smooth functions on R that decay like (
√
1 + x2)β
as |x| → ∞, for some β < 0. This algebra is fully characterized in [9]).
We then show that our functional calculus coincides with C0-functional
calculus for an unbounded operator acting on a Hilbert space.
Mathematics Subject Classification: 47A60