On the generation of Arveson weakly continuous semigroups
| hal.structure.identifier | Institut de Mathématiques de Bordeaux [IMB] | |
| dc.contributor.author | ESTERLE, Jean | |
| dc.date.accessioned | 2024-04-04T03:08:51Z | |
| dc.date.available | 2024-04-04T03:08:51Z | |
| dc.date.created | 2017-09 | |
| dc.identifier.uri | https://oskar-bordeaux.fr/handle/20.500.12278/193563 | |
| dc.description.abstractEn | We consider here one-parameter semigroups T = (T (t))t>0 of bounded operators on a Banach space X which are weakly continuous in the sense of Arveson. For such a semigroup T denote by Mω T the convolution algebra consisting in those measures µ on (0, +∞) such that R +∞ 0 T (t)d|µ|(t) < +∞. The Pettis integral R +∞ 0 T (t)dµ(t) defines for µ ∈ Mω T a bounded operator φ T (µ) on X. Identifying the space L 1 ω T of (classes of) measurable functions f satisfying R +∞ 0 |f (t)T (t)dt < +∞ to a closed subspace Mω T in the usual way, we define the Arveson ideal I T of the semigroup to be the closure in B(X) of φ T (L 1 ω T). Using a variant of a procedure introduced a long time ago by the author we introduce a dense ideal U T of I T , which is a Banach algebra with respect to a suitable norm .U T , such that lim sup t→0 + T (t) B(U T) < +∞. The normalized Arveson ideal J T is the closure of I T in B(U T). The Banach algebra J T has a sequential approximate identity and is isometrically isomorphic to a closed ideal of its multiplier algebra M(J T). The Banach algebras U T , I T and J T are "similar", and the map S u/v → S au/av defines when a generates a dense principal ideal of U T a pseudobounded isomorphism from the alge-bre QM(J T) of quasimultipliers on J T onto the quasimultipliers algebras QM(U T) and QM(I T). We define the generator A T of the semigroup T to be a quasimul-tiplier on I T , or ,equivalently, on J T. Every character χ on I T has an extensioñ χ to QM(I T). Let Resar(A T) be the complement of the set { ˜ χ(A T)} χ∈ d I T. The quasimultiplier A − µI has an inverse belonging to J T for µ ∈ Resar(A T), which allows to consider this inverse as a "regu-lar" quasimultiplier on the Arveson ideal I T. The usual resolvent formula holds in this context for Re(µ) > limt→+∞ logT (t) t. Set Π + α := {z ∈ C | Re(z) > α}. We revisit the functional calculus associated to the generator A T by defining F (−A T) ∈ J T by a Cauchy integral when F belongs to the Hardy space H 1 (Π + α) for some α < − limt→+∞ logT (t)| t. We then define F (−A T) as a quasimultiplier on J T and I T when F belongs to the Smirnov class on Π + α , and F (−A T) is a regular quasimultiplier on J T and I T if F is bounded on Π + α. If F (z) = e −zt for some t > 0, then F (−A T) = T (t), and if F (z) = −z, we indeed have F (−A T) = A T . | |
| dc.language.iso | en | |
| dc.subject.en | Arveson pairs | |
| dc.subject.en | Arveson spec- | |
| dc.subject.en | trum | |
| dc.subject.en | Pettis integral | |
| dc.subject.en | infinitesimal generator | |
| dc.subject.en | resolvent | |
| dc.subject.en | Laplace transform | |
| dc.subject.en | holo- | |
| dc.subject.en | morphic functional calculus | |
| dc.subject.en | AMS classification: Primary 47A16 | |
| dc.subject.en | Secondary 47D03 | |
| dc.subject.en | 46J40 | |
| dc.subject.en | 46H20 | |
| dc.subject.en | 1 | |
| dc.subject.en | keywords: semigroup of bounded operators | |
| dc.title.en | On the generation of Arveson weakly continuous semigroups | |
| dc.type | Document de travail - Pré-publication | |
| dc.subject.hal | Mathématiques [math]/Analyse fonctionnelle [math.FA] | |
| dc.identifier.arxiv | 1709.05218 | |
| 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-01587949 | |
| hal.version | 1 | |
| hal.origin.link | https://hal.archives-ouvertes.fr//hal-01587949v1 | |
| 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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