Bergman Kernel, Szego Kernel and Dirichlet Integral

D. A. Hejhal showed the positive definiteness of the matrix H = (h(jk)) appearing in Schiffer's identity K-B = 4 pi K-S(2) + Sigma (j,k) h (jk) v j (v(k)) over bar, where K-B and K-S are, respectively, the Bergman and the Szego kernel on a planar regular region. We give several conditions equivalent to Hejhal's theorem by means of an integral transform whose kernel is the product of two Szego kernels. In particular, we show that the positive definiteness of thematrix H follows from the contractivity of the above integral transform together with the equality condition fornorms.