Spectralproperties of the Cauchy operator and its product with Bergman's projection on a bounded domain

Abstract

In this paper exact asymptotic formulae are found for singular values of the Cauchy operator and the logarithmic potential type operator (on a bounded domain), as well as their products with Bergman's projection. It is shown that these spectral characteristics detect geometric properties of a domain $\Omega$ (area and the length of the boundary). The hypothesis “can we hear the shape of a drum”, from a paper by J.M. Anderson, D. Khavinson, and V. Lomonosov [‘Spectral propertiesof some integral operators arising in potential theory’, {\em Quart.\ J. Math.\ Oxford} (2) 43 (1992) 387-407], is correct in the above sense.

1991 Mathematics Subject Classification: 47B10.