Invariant submodules of modular operators and Lomonosov type theorem for Hilbert C*-modules

TL;DR

This paper extends the concept of invariant submodules to Hilbert C*-modules, providing new insights into operator equations and establishing a Lomonosov type theorem for compact operators in this setting.

Contribution

It introduces invariant submodules in Hilbert C*-modules and proves a Lomonosov type theorem for compact operators on these modules.

Findings

Representation of solutions to operator equations using invariant submodules

Lomonosov type theorem for compact operators on Hilbert C*-modules

Simplified results for finite dimensional and compact operator C*-algebras

Abstract

In this paper, we introduce the notion of invariant submodule in the theory of Hilbert C*-modules and study some basic properties of bounded adjointable operators and their generalized inverses which have nontrivial invariant submodules. We demonstrate the representation of the solution set of an operator equation on Hilbert C*-modules by taking advantage of invariant submodules. In particular, we consider the special cases of finite dimensional C*-algebras and C*-algebras of compact operators as the underling C*-algebra to simplify our results, and obtain a Lomonosov type theorem for compact operators on some Hilbert C*-modules.