Symmetric Kernels

Abstract

A kernel K (s, t) is symmetric (or complex symmetric or Hermitian) if K(s, t) = K*(t, s), (7.1.1) where the asterisk denotes the complex conjugate. In the case of a real kernel, the symmetry reduces to the equality K(s, t) = K(t, s). (7.1.2) We have seen in the previous two chapters that the integral equations with symmetric kernels are of frequent occurrence in the formulation of physically motivated problems. We claim that if a kernel is symmetric, then all its iterated kernels are also symmetric. Indeed, Kz(s, t) = J K(s, x)K(x, t)dx = j K*(t, x)K*(x, s)dx = K;(t, s). Again, if Kn(s, t) is symmetric, then the recursion relation gives Kn+l(s, t) = J K(s, x)Kn(X, t)dx = J K;(t,x)K*(x,s)dx=K:+l(t,s). (7.1.3) The proof of our claim follows by induction. Note that the trace K (s, s) of a symmetric kernel is always real because K (s, s) = K* (s, s). Similarly, the traces of all iterates are also real. Complex Hilbert space We present here a brief review of the properties of the complex Hilbert space £ 2 (a, b), which is needed in the sequel. This discussion is valid for real Cz-space as a special case. A linear space of infinite dimension with inner product (or scalar product) (x, y) which is a complex number satisfying (a) the definiteness axiom (x, x) > 0 for x # 0; (b) the linearity axiom (ax1 +fJxz, y) = a(x1, y) + R. P. Kanwal, Linear Integral Equations