We find all the spherical functions on the space of infinite dimensional Hermitian matrices with reels, complexes or quaternions coefficients witch are invariantes by the action of the infinite unitary group.In the first chapter we give sum classical results demonstrate by J. Faraut and A. Koranyi.In the second chapter we give a application of the theorems of Minlos and Poincaré to find the limits of some orbitale integrale.The third chapter is the means result. We find a formula for the spherical functions and we prove that the set of the spherical functions is paramatrised by some infinite set.In the chapter number for we give the Bochner invariant theorem and some ather complements about positive definite function.In the last chapter we concentrate to give a Lévy-Khinchine representation of all invariantes negative definite functions on the space of hermitiens and Hilbert-Schmidt type matrices.
