The relationships between Invertible Module Maps X and X_z

TL;DR

This paper investigates the conditions under which invertibility of module maps between quotient modules at each point implies the invertibility of the original module map, focusing on Hilbert modules and their quotient structures.

Contribution

It establishes a criterion linking pointwise invertibility of quotient module maps to the invertibility of the original module map in the context of quasi-free Hilbert modules.

Findings

Invertibility of $X_z$ for all $z$ implies invertibility of $X$ under certain conditions.

Provides a specific condition for module maps between quasi-free Hilbert modules.

Connects local invertibility properties to global invertibility of module maps.

Abstract

If $R$ and $M$ are Hilbert modules (in the sense of R. G. Douglas and V. I. Paulsen), we study the relationship between invertible module maps $X:R\to{M}$ and $X_{z}:R/R_{z}\to{M/M_{z}}$. In particular, for quasi-free Hilbert modules $R$ and $M$, we provide a condition of a module map $X:R\to{M}$, such that if $X_{z}:R/R_{z}\to{M/M_{z}}$ is invertible for every $z$ in a domain $\Omega$ in the complex plane, then $X$ is also invertible.