 
Summary: HOLOMORPHIC FUNCTIONAL CALCULI AND SUMS OF
COMMUTING OPERATORS
DAVID ALBRECHT, EDWIN FRANKS AND ALAN M c INTOSH
Abstract. 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], maccretive operators and msectorial operators [Kat]. In the
paper [M c ], one of the authors developed a functional calculus for operators of
type ! on Hilbert spaces which was extended to include more general Banach
spaces in [CDM c Y]. This calculus provides a unified theory for which fractional
powers, exponentials, logarithms, imaginary powers, and other functions, includ
