Spectral Properties of the Cauchy Operator and Its Product with Bergman's Projection on a Bounded Domain

1998/05/01 by MR Dostanić, M. R. Dostanić · 1 citation
Mathematics · Computer Science · #Holomorphic and Operator Theory #Algebraic and Geometric Analysis #Advanced Mathematical Modeling in Engineering

paper · doi:10.1112/s0024611598000367

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 Ω (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 properties of some integral operators arising in potential theory’, Quart. J. Math. Oxford (2) 43 (1992) 387-407], is correct in the above sense. 1991 Mathematics Subject Classification: 47B10.

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