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.