A Functional Calculus for (α, α+ 1)− type R Operators

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