Spectral representation of the weighted Laplace transform

We find the spectral representation of the selfadjoint operators T T f (lambda) := integral(infinity)(0) K (lambda t) f (t) dt, in L-2(]0, infinity[), where 0 <= K is an element of L-loc(1)(0, infinity). More precisely (see Theorem 4.1) for these operators which include the Laplace transform as a special case, the spectrum of T is a compact interval [-kappa, kappa], and we find explicitly a unitary operator U : L-2(]0, infinity[) -> L-2(R) and a continuous real function alpha on R such that UTU-1 is the operator of multiplication by alpha.