1. Weyl transforms and the generalized Hermite operator.The abstract Cauchy problem (ACP) for the abstract Hermite operator. Viorel Catana University Politehnica of Bucharest Department of Mathematics I catana_viorel@yahoo.co.uk The 10th French-Romanian Conference on Applied Mathematics Poitiers (Futuroscope ),France, August 26-31, 2010

2. Topics Introduction Introduction Basic definitions and notations Basic definitions and notations Preliminary results Preliminary results The generalized Hermite semigroup The generalized Hermite semigroup The inverse of the generalized Hermite operator The inverse of the generalized Hermite operator The heat kernel of the generalized Hermite operator The heat kernel of the generalized Hermite operator The Green function of the generalized Hermite operator The Green function of the generalized Hermite operator Global hypoellipticity Global hypoellipticity Ultracontractivity and Hypercontractivity Ultracontractivity and Hypercontractivity The abstract Hermite semigroup The abstract Hermite semigroup The abstract Cauchy problem (ACP) for the abstract Hermite operator The abstract Cauchy problem (ACP) for the abstract Hermite operator

3. Introduction

4. References

7. The twisted Laplacian

8. Wong’s research about twisted Laplacian

9. Basic definitions & notations

10. The generalized Hermite operator

14. Basic transformations

15. The generalized Hermite functions

16. Preliminary results

17. Main results concerning the Hermite semigroup

19. For the proof of Theorem 6 we need the following two results (see Chapter 17 of the book [17] and Theorem 15.4 and Theorem 17.1 in the book [16] by Wong).

20. The inverse of the generalized Hermite operator

22. The heat kernel

23. Preliminary result

29. The Green function of the generalized Hermite operator

31. Global hypoellipticity

34. Ultracontractivity and Hypercontractivity

37. The abstract Cauchy problem (ACP) for the abstract Hermite operator

39. The abstract Hermite operator

40. The semigroup of the abstract Hermite operator

43. An estimate of the abstract Hermite semigroup

44. The (ACP) for the abstract Hermite operator

46. Merci pour votre aimable attention! Thank you for your kind attention!