The angle of an operator and range and kernel complementarity

TL;DR

This paper investigates the geometric relationship between the angle of a bounded linear operator and the complementarity of its range and kernel, providing new characterizations and conditions in Banach and Hilbert spaces.

Contribution

It establishes a link between the operator's angle and the complementarity of its range and kernel, offering new proofs and conditions involving the numerical range boundary.

Findings

Operators with angle less than π have complementary range and kernel.

In finite dimensions, the property characterizes certain operators up to rotations.

A sufficient condition involving the numerical range boundary is provided for Hilbert space operators.

Abstract

We show that if the angle of a bounded linear operator on a Banach space, with closed range and closed sum of its range and kernel, is less than $\pi$, then its range and kernel are complementary. In finite dimensions and up to rotations this simple geometric property characterizes operators for which the above complementarity holds. Applying our result we get simple proofs concerning eigenvalues lying in the boundary of the numerical range. Concluding, for an operator on a Hilbert space, we present a sufficient condition, involving the distance of the boundary of the numerical range from the origin, for its range and kernel to be complementary.