Bilinear expansions of the kernels of some nonselfadjoint integral operators

Abstract

Let $H$ and $S$ be integral operators on $L^2(0,1)$ with continuous kernels. Suppose that $H>0$ and let $A=H(I+S)$. It is shown that if the (nonselfadjoint) operator $S$ is small in a certain sense with respect to $H$, then the corressponding Fourier series of functions from $R(A)$ (or $R(A^{\ast })$) converges uniformly on $[0,1]$.