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Invariant measure for some differential operators and unitarizing measure for the representation of a Lie group. Examples in finite dimension

100%

Airault H. , Ouerdiane H.

Banach Center Publications

2011

tom 96

nr 1

9-34

Consider a Lie group with a unitary representation into a space of holomorphic functions defined on a domain 𝓓 of ℂ and in L²(μ), the measure μ being the unitarizing measure of the representation. On finite-dimensional examples, we show that this unitarizing measure is also the invariant measure for some differential operators on 𝓓. We calculate these operators and we develop the concepts of unitarizing measure and invariant measure for an OU operator (differential operator associated to the representation) in the following elementary cases: A) The commutative groups (ℝ,+) and (ℝ* = ℝ-0,×). B) The multiplicative group M of 2×2 complex invertible matrices and some subgroups of M. C) The three-dimensional Heisenberg group.

Infinite dimensional Gegenbauer functionals

80%

Barhoumi A. , Ouerdiane H. , Riahi A.

Banach Center Publications

2007

tom 78

nr 1

35-45

he paper is devoted to investigation of Gegenbauer white noise functionals. A particular attention is paid to the construction of the infinite dimensional Gegenbauer white noise measure $𝒢_{β}$, via the Bochner-Minlos theorem, on a suitable nuclear triple. Then we give the chaos decomposition of the L²-space with respect to the measure $𝒢_{β}$ by using the so-called β-type Wick product.