Spectrum and resolvent of a partial integral operator

摘要

We consider the partial integral operator h→w(x,y)h(x,y) + ∫(K(x,y,s)h(x,s)ds(0≤s≤φ(x))(a≤x≤b;0≤y≤φ(x)) where w(., .), K(., ., .), φ are given functions. Norm estimates for the resolvent and bounds for the spectrum are established. Our main tool is the direct integral of Hilbert spaces and recent estimates for norms of the resolvents of compact operators and operators with the compact Hermitian components.