A geometric characterization of range-kernel complementarity

TL;DR

This paper characterizes when a bounded linear operator on a Banach space has a complementary range and kernel using a geometric condition involving its generalized amplitude, and applies this to analyze the convergence of operator iterations.

Contribution

It provides a geometric criterion for range-kernel complementarity based on the operator's generalized amplitude, linking geometric properties to operator theory.

Findings

Range-kernel complementarity occurs iff the generalized amplitude is less than π.

Application to strong convergence of bounded linear operator iterations.

Provides a geometric perspective on operator properties.

Abstract

We show that a bounded linear operator on a Banach space with closed range has range-kernel complementarity if and only if its generalized amplitude is less than $\pi$. An application to the strong convergence of the iterations of a bounded linear operator is also given.