Spectral properties of the Cauchy transform on L2(C, e−|z|2 λ(z))

Let hm,p(z), (m,p) ∈ Z+ × Z+, be the Landau orthogonal basis of the Hilbert space on L2(C, e−|z|2 dλ(z)) where λ(z) is the usual Lebesgue measure on the complex plane. In this paper we give some spectral properties of the Cauchy transform on the orthogonal complement of Bargmann space Λ0(C) in L2(C, e−|z|2 dλ(z)). In particular for m fixed, we consider the orthogonal projection operator on the Hilbert subspace spanned by hm,p(z), p = 0, 1, 2, . . . , and we give explicitly the sequence of singular values of its composition with the Cauchy transform in L2(C, e−|z|2 dλ(z)). As application of these of the Cauchy transform we get some identities for special functions which could be of independent interest.  2005 Elsevier Inc. All rights reserved