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Proceedings of the London Mathematical Society Advance Access originally published online on November 27, 2006
Proceedings of the London Mathematical Society 2007 94(1):181; doi:10.1112/plms/pdl013
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© 2006 London Mathematical Society

Invariant Gaussian measures for operators on Banach spaces and linear dynamics

Frédéric Bayart1 and Sophie Grivaux2

1 Laboratoire Bordelais d’Analyse et de Géométrie
UMR 5467
Université Bordeaux 1
351, Cours de la Libération
33405 Talence cedex
France
Frederic.Bayart{at}math.u-bordeaux1.fr
2 Laboratoire Paul Painlevé
UMR 8524
Université des Sciences et Technologies de Lille Cité Scientifique
59655 Villeneuve d’Ascq cedex
France
grivaux{at}math.univ-lille1.fr

Received 23 September 2005. Revision received 6 April 2006.

We give conditions for an operator T on a complex separable Banach space X with sufficiently many eigenvectors associated to eigenvalues of modulus 1 to admit a non-degenerate invariant Gaussian measure with respect to which it is weak-mixing. The existence of such a measure depends on the geometry of the Banach space and on the possibility of parametrizing the T-eigenvector fields of T in a regular way. We also investigate the connection with frequent hypercyclicity.


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