A holomorphic functional calculus for finite families of commuting semigroups
| hal.structure.identifier | Institut de Mathématiques de Bordeaux [IMB] | |
| dc.contributor.author | ESTERLE, Jean | |
| dc.date.accessioned | 2024-04-04T03:02:37Z | |
| dc.date.available | 2024-04-04T03:02:37Z | |
| dc.identifier.uri | https://oskar-bordeaux.fr/handle/20.500.12278/193004 | |
| dc.description.abstractEn | Let A be a commutative Banach algebra such that uA = {0} for u ∈ A \ {0} which possesses dense principal ideals. The purpose of the paper is to give a general framework to define F (−λ1∆T 1 ,. .. , −λ k ∆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 (−λ1∆T 1 ,. .. , −λ k ∆T k), where ∆T 1 ,. .. , ∆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 ∪t>0T (te ia)A is dense in A, where a ∈ R. (2) The class of semigroups T = (T (ζ)) ζ∈S a,b holomorphic on an open sector S a,b such that T (ζ)A is dense in A for some, or equivalently for all ζ ∈ 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 ζ in the Arveson resolvent set σar(∆T) the resolvent (∆T − ζI) −1 will be defined as a regular quasimultiplier on A, i.e. a quasimultiplier S on A such that sup n≥1 λ n S n u < +∞ for some λ > 0 and some u generating a dense ideal of A and belonging to the intersection of the domains of S n , n ≥ 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→0 + T (te ia) M(B) < +∞ if T belongs to the class (1), and for which lim sup ζ→0 ζ∈S α,β T (ζ) < +∞ for all pairs (α, β) such that a < α < β < b if T belongs to the class (2). Iterating this procedure this allows to consider (λj∆T j + ζI) −1 as an element of M(B) for ζ ∈ Resar(−λj∆T j), the "Arveson resolvent set " of −λj∆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 ≤ bj ≤ aj + π, and denote by M a,b the set of families (α, β) = (α1, β1),. .. , (α k , β k) such that 1 | |
| dc.language.iso | en | |
| dc.title.en | A holomorphic functional calculus for finite families of commuting semigroups | |
| dc.type | Document de travail - Pré-publication | |
| dc.subject.hal | Mathématiques [math]/Analyse fonctionnelle [math.FA] | |
| dc.subject.hal | Sciences de l'ingénieur [physics]/Autre | |
| dc.identifier.arxiv | 1901.00293 | |
| bordeaux.hal.laboratories | Institut de Mathématiques de Bordeaux (IMB) - UMR 5251 | * |
| bordeaux.institution | Université de Bordeaux | |
| bordeaux.institution | Bordeaux INP | |
| bordeaux.institution | CNRS | |
| hal.identifier | hal-01966621 | |
| hal.version | 1 | |
| hal.origin.link | https://hal.archives-ouvertes.fr//hal-01966621v1 | |
| bordeaux.COinS | ctx_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.au=ESTERLE,%20Jean&rft.genre=preprint |
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