Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A holomorphic functional calculus for finite families of commuting
  semigroups   Let A be a commutative Banach algebra such that uA = {0} for u $\in$ A \ {0}
which possesses dense principal ideals. The purpose of the paper is to give a
general framework to define F (--$\lambda$1$\Delta$T 1 ,. .. , --$\lambda$ k
$\Delta$T k) where F belongs to a natural class of holomorphic functions
defined on suitable open subsets of C k containing the "Arveson spectrum" of
(--$\lambda$1$\Delta$T 1 ,. .. , --$\lambda$ k $\Delta$T k), where $\Delta$T 1
,. .. , $\Delta$T k are the infinitesimal generators of commuting one-parameter
semigroups of multipliers on A belonging to one of the following classes (1)
The class of strongly continous semigroups T = (T (te ia)t>0 such that
$\cup$t>0T (te ia)A is dense in A, where a $\in$ R. (2) The class of semigroups
T = (T ($\zeta$)) $\zeta$$\in$S a,b holomorphic on an open sector S a,b such
that T ($\zeta$)A is dense in A for some, or equivalently for all $\zeta$ $\in$
S a,b. We use the notion of quasimultiplier, introduced in 1981 by the author
at the Long Beach Conference on Banach algebras: the generators of the
semigroups under consideration will be defined as quasimultipliers on A, and
for $\zeta$ in the Arveson resolvent set $\sigma$ar($\Delta$T) the resolvent
($\Delta$T -- $\zeta$I) --1 will be defined as a regular quasimultiplier on A,
i.e. a quasimultiplier S on A such that sup n$\ge$1 $\lambda$ n S n u <
+$\infty$ for some $\lambda$ > 0 and some u generating a dense ideal of A and
belonging to the intersection of the domains of S n , n $\ge$ 1. The first step
consists in "normalizing" the Banach algebra A, i.e. continuously embedding A
in a Banach algebra B having the same quasi-multiplier algebra as A but for
which lim sup t$\rightarrow$0 + T (te ia) M(B) < +$\infty$ if T belongs to the
class (1), and for which lim sup $\zeta$$\rightarrow$0 $\zeta$$\in$S
$\alpha$,$\beta$ T ($\zeta$) < +$\infty$ for all pairs ($\alpha$, $\beta$) such
that a < $\alpha$ < $\beta$ < b if T belongs to the class (2). Iterating this
procedure this allows to consider ($\lambda$j$\Delta$T j + $\zeta$I) --1 as an
element of M(B) for $\zeta$ $\in$ Resar(--$\lambda$j$\Delta$T j), the "Arveson
resolvent set " of --$\lambda$j$\Delta$T j , and to use the standard integral
'resolvent formula' even if the given semigroups are not bounded near the
origin. A first approach to the functional calculus involves the dual G a,b of
an algebra of fast decreasing functions, described in Appendix 2. Let a = (a1,.
.. , a k), b = (b1,. .. , b k), with aj $\le$ bj $\le$ aj + $\pi$, and denote
by M a,b the set of families ($\alpha$, $\beta$) = ($\alpha$1, $\beta$1),. .. ,
($\alpha$ k , $\beta$ k) such that 1

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