On the eigenvalues of a certain integral equation

It is shown that the integral equation \[ \int_0^\infty {f(t)} \frac{{\sin (x - t)}} {{\pi (x - t)}}dt = \lambda f(x)\quad (x > 0)\] has a solution $f$ for any (complex) $\lambda $, excluding the real numbers $\lambda \leq 0$, $\lambda > 1$. The closure of the set of eigenvalues, i.e., the spectrum of the integral operator (over all functions in its domain) is then the entire complex plane.