Spectral properties of an operator of Riesz potential type and its product with the Bergman projection on a bounded domain (Q2710723)

Let \(\Omega\) be a bounded simply connected domain in \({\mathbb C}\). By \(A\) the operator of the Riesz potential type on \(L^2(\Omega)\) is defined. Let \(L_a^2(\Omega)\subset L^2(\Omega)\) be the Bergman space, and \(P\) the orthogonal projector from \(L^2(\Omega)\) onto \(L_a^2(\Omega)\) (the Bergman projector). An exact asymptotic formula for the singular values of the product \(AP\) of an operator of Riesz potential type and the Bergman projection on a bounded domain is obtained. Moreover, the author shows that these singular values determine the length of the boundary of the domain. These results supplement the well-known fact that the spectrum of the operator of Riesz potential type determines the area of the domain.