Holomorphic functional calculi and sums of commuting opertors

Let S and T be commuting operators of type ω and type ω̄ in a Banach space X. Then the pair has a joint holomorphic functional calculus in the sense that it is possible to define operators f(S,T) in a consistent manner, when f is a suitable holomorphic function defined on a product of sectors. In particular, this gives a way to define the sum S + T when ω + ω̄ < π. We show that this operator is always of type μ where μ = max{ω, ω̄}. We explore when bounds on the individual functional calculi of S and T imply bounds on the functional calculus of the pair (S, T), and some implications for the regularity problem of when ∥(S + T)u∥ is equivalent to ∥Su∥ + ∥Tu∥.