Eigenvalue estimates of positive integral operators with analytic kernels
Suppose that \(\Omega\) is a simply connected domain with an infinite complement, \(\Delta\) is the unit disc, \(\varphi : \Omega \rightarrow \Delta \) is a conformal map, \[ K_{\Omega} (z, \zeta)= \frac{\left(\varphi'(\zeta)\right)^{\frac{1}{2}} \overline{\left(\varphi'(z)\right)^{\frac{1}{2}}}}{1-\varphi(\zeta)\overline{\varphi(z)}}. \] Suppose that \(J\) is a closed real interval such that \(J \subset \Omega\) and \[ (T_{\Omega}f)(s) = \int_{J} K_{\Omega}(s, t) f(t)dt, \;s\in J. \] The operator \(T_{\Omega}\) is compact and positive (\((T_{\Omega} f, f) \geq 0\)) in the space \(L_2(J)\). Such operators studied by \textit{G. Little} in his papers [Math. Proc. Camb. Philos. Soc. 91, 267--284 (1982; Zbl 0489.45001), Proc. Lond. Math. Soc., III. Ser. 62, No.~2, 403--426 (1991; Zbl 0682.45004)]. Let \(C=(C_1, \dots , C_N) \) be an \(N\)-tuple of conic sections. Let \(D\) be a complementary domain of \(\bigcup_{i=1}^N C_i\), and suppose that \(D\) is simply connected. Let \(D_i\) be the complementary domain of \(C_i\) which contains \(D\). The main result of the author states that there exists some constant \(m>0\) such that \( m T_D \leq T_{D_1} +\cdots + T_{D_N}\).
scientific article; zbMATH DE number 5810051