A Riesz-Fredholm type theorem on certain Hilbert C*-modules

Let $C$ be compact modular operator on a Hilbert C*-module $E$ satisfying property $\mathbb{[H]}$ [{\it J. Math. Phys.} {\bf 49} (2008), 033519], and let $L :=I-C$. We prove the existence of a unique natural number $r$ for which $L^r$ is an EP operator on $E$. Moreover, we show that the kernel of $L^r$ is a finitely generated submodule of $E$ and that $E$ admits the decomposition $E=Ker(L^r) \oplus Ran(L^r)$. These results provide a framework for analyzing the solvability of the equation $x-Cx=f$ on $E$.