  The reduced (in the angular coordinate $\phi$) wave equation and Klein-Gordon
equation are considered on a Kerr background and in the framework of
$C^{0}$-semigroup theory. Each equation is shown to have a well-posed initial
value problem,i.e., to have a unique solution depending continuously on the
data. Further, it is shown that the spectrum of the semigroup's generator
coincides with the spectrum of an operator polynomial whose coefficients can be
read off from the equation. In this way the problem of deciding stability is
reduced to a spectral problem and a mathematical basis is provided for mode
considerations. For the wave equation it is shown that the resolvent of the
semigroup's generator and the corresponding Green's functions can be computed
using spheroidal functions. It is to be expected that, analogous to the case of
a Schwarzschild background, the quasinormal frequencies of the Kerr black hole
appear as {\it resonances}, i.e., poles of the analytic continuation of this
resolvent. Finally, stability of the background with respect to reduced massive
perturbations is proven for large enough masses.
