Holomorphic Functional Calculi And Sums Of Commuting Operators

. Let S and T be commuting operators of type ! and type $ 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 !+$ ! ß. We show that this operator is always of type ¯ where ¯ = maxf!; $g. 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 k(S + T )uk is equivalent to kSuk + kTuk. 1. Introduction The class of operators of type ! is important in functional analysis and partial differential equations. Well known examples include elliptic operators on various domains [GS, See], m-accretive operators and m-sectorial operators [Kat]. In the paper [M c ], one of the authors developed a functional calculus for ope..