A Riesz-Fredholm Type Theorem on Hilbert C*-Modules Satisfying Property \mathbb{ \left[H\right]}

Consider a compact modular operator C on a Hilbert C*-module E that fulfills property \mathbb{ \left[H\right]} [{\em J. Math. Phys.}, {۴۹} (۲۰۰۸), ۰۳۳۵۱۹] and define 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\left(L^r \right) \oplus ran \left(L^r\right). These results provide a framework for analyzing the solvability of the equation x-Cx=f on E.

کلیدواژه ها:

Hilbert C*-module ، Compact operator ، Moore-Penrose inverse ، EP operator ، Finitely generated submodule