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

TL;DR

The paper extends the classical Riesz–Fredholm framework to Hilbert C*-modules satisfying property $\mathcal{[H]}$ by analyzing the equation $x - Cx = f$ with a compact operator $C$. It proves that there exists a unique $r \ge 0$ for which $L^r = (I - C)^r$ is EP with closed range, and that $E$ decomposes as $E = \ker(L^r) \oplus \mathrm{Ran}(L^r)$ with $\ker(L^r)$ finitely generated. This yields solvability criteria for the equation, a matrix-type decomposition on the range, and corollaries detailing cases $r=0$ and $r>0$, as well as a potential generalization to $L = \lambda I - C$. The results provide a robust structure for studying linear equations involving compact modular operators on Hilbert C*-modules, extending the Riesz–Fredholm paradigm beyond classical Hilbert spaces.

Abstract

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 $E$.

Theorems & Definitions (25)

• Definition 2.1
• Lemma 2.2
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Lemma 3.1
• Lemma 3.2
• proof
• Lemma 3.3