SPECTRA OF UNITARY INTEGRAL OPERATORS ON L² (ℝ) WITH KERNELS k(xy)

Unitary integral transforms play an important role in mathematical physics. A primary example is the Fourier transform whose kernel is of the form k(x,y) = k(xy), i.e., of the product type. Here we consider the determination of spectrum for such unitary operators as the issue is important in the solvability of the corresponding inhomogeneous Fredholm integral equation of the second kind. A Main Theorem is proven that characterizes the spectral set. Properties of eigenfunctions and eigenspace dimensions are further derived as consequences of the Main Theorem. Concrete examples are also offered as applications.