Spectral analysis of finite convolution operators

Abstract:

In this paper the similarity problem for operators of the form $( \ast )\;T:f(x) \to \smallint _0^xk(x - t)f(t)dt$ on ${L^2}(0,1)$ is studied. Let $K(z) = \smallint _0^1\;k(t){e^{itz}}dt$. A function $C(z)$ is called a symbol for T if $C(z)$ can be written in the form $C(z) = K(z) + {e^{iz}}G(z)$, where $G(z)$ is a function bounded and analytic in a half plane $y > \delta$, for some real number $\delta$. Under suitable restrictions, it is shown that two operators of the form $( \ast )$ will be similar if they possess symbols which are asymptotically close together as $z \to \infty$ in some half plane $y > \delta$.