Commutator conditions implying the convergence of the Lie–Trotter products

Kühnemund, Franziska; Wacker, Markus

doi:10.1090/S0002-9939-01-06034-8

Commutator conditions implying the convergence of the Lie-Trotter products

Authors: Franziska Kühnemund and Markus Wacker
Journal: Proc. Amer. Math. Soc. 129 (2001), 3569-3582
MSC (2000): Primary 34G10, 35K15, 47D06
Published electronically: April 25, 2001
MathSciNet review: 1860489
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Abstract:

In this paper we investigate commutator conditions for two strongly continuous semigroups ${(T(t))_{t\geq 0}\,}$ and ${(S(t))_{t\geq 0}\,}$ on a Banach space implying the convergence of the Lie-Trotter products $[T(\tfrac{t}{n})S(\tfrac{t}{n})]^n$ . The results are then applied to various examples and, in particular, to the Ornstein-Uhlenbeck operator.

1. J. Bass, Cours de mathématiques. Tomes I, II, Quatrième édition revue et corrigée, Masson et Cie, Éditeurs, Paris, 1968 (French). MR 0224402
2. E. Barucci, S. Polidoro, V. Vespri, Some results on partial differential equations and Asian options, to appear.
3. Piermarco Cannarsa and Giuseppe Da Prato, On a functional analysis approach to parabolic equations in infinite dimensions, J. Funct. Anal. 118 (1993), no. 1, 22–42. MR 1245596, 10.1006/jfan.1993.1137
4. Paul R. Chernoff, Product formulas, nonlinear semigroups, and addition of unbounded operators, American Mathematical Society, Providence, R. I., 1974. Memoirs of the American Mathematical Society, No. 140. MR 0417851
5. Giuseppe Da Prato and Alessandra Lunardi, On the Ornstein-Uhlenbeck operator in spaces of continuous functions, J. Funct. Anal. 131 (1995), no. 1, 94–114. MR 1343161, 10.1006/jfan.1995.1084
6. Klaus-Jochen Engel and Rainer Nagel, One-parameter semigroups for linear evolution equations, Graduate Texts in Mathematics, vol. 194, Springer-Verlag, New York, 2000. With contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt. MR 1721989
7. Palle E. T. Jorgensen and Robert T. Moore, Operator commutation relations, Mathematics and its Applications, D. Reidel Publishing Co., Dordrecht, 1984. Commutation relations for operators, semigroups, and resolvents with applications to mathematical physics and representations of Lie groups. MR 746138
8. F. Kühnemund and M. Wacker, The Lie-Trotter product formula does not hold for arbitrary sums of generators, to appear in Semigroup Forum.
9. Sylvie Monniaux and Jan Prüss, A theorem of the Dore-Venni type for noncommuting operators, Trans. Amer. Math. Soc. 349 (1997), no. 12, 4787–4814. MR 1433125, 10.1090/S0002-9947-97-01997-1
10. Michael Reed and Barry Simon, Methods of modern mathematical physics. I. Functional analysis, Academic Press, New York-London, 1972. MR 0493419
11. Walter Rudin, Functional analysis, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. McGraw-Hill Series in Higher Mathematics. MR 0365062
12. Hiroyuki Suzuki, On the commutation relation 𝐴𝐵-𝐵𝐴=𝐶, Proc. Japan Acad. 47 (1971), 173–176. MR 0291878
13. H. F. Trotter, On the product of semi-groups of operators, Proc. Amer. Math. Soc. 10 (1959), 545–551. MR 0108732, 10.1090/S0002-9939-1959-0108732-6
14. Kôsaku Yosida, Functional analysis, 4th ed., Springer-Verlag, New York-Heidelberg, 1974. Die Grundlehren der mathematischen Wissenschaften, Band 123. MR 0350358

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Additional Information

Franziska Kühnemund
Affiliation: Mathematisches Institut, Auf der Morgenstelle 10, D-72076 Tübingen, Germany
Email: frku@michelangelo.mathematik.uni-tuebingen.de

Markus Wacker
Affiliation: Mathematisches Institut, Auf der Morgenstelle 10, D-72076 Tübingen, Germany
Email: mawa@michelangelo.mathematik.uni-tuebingen.de

DOI: http://dx.doi.org/10.1090/S0002-9939-01-06034-8
Keywords: Strongly continuous semigroups, Trotter--Lie product formula, commutator conditions, Weyl relation, Ornstein--Uhlenbeck semigroup
Received by editor(s): December 16, 1999
Received by editor(s) in revised form: April 14, 2000
Published electronically: April 25, 2001
Additional Notes: The authors thank Giorgio Metafune, Rainer Nagel, Abdelaziz Rhandi and Roland Schnaubelt for helpful discussions.
Communicated by: David R. Larson
Article copyright: © Copyright 2001 American Mathematical Society