Spectral estimates of Cauchy's operator on Bergman space of harmonic functions

In this paper we investigate the asymptotic behaviour of singular numbers of the operator PhCPhPhCPh, where C   is Cauchy's operator and PhPh is the orthogonal projection from L2(D)L2(D) onto harmonic function subspace. We prove that sn(PhCPh)=O(1n), as n→+∞n→+∞. Moreover, we find the upper and lower asymptotic estimates, π−1⩽limn→+∞⁡nsn(PhCPh)⩽π−1(35+2126)+76.