# The angle along a curve and range-kernel complementarity

## TL;DR

This paper introduces a new concept of the angle of a bounded linear operator along a path to characterize when the operator's range and kernel are complementary, linking geometric properties to operator theory.

## Contribution

It defines the angle along a curve for operators and uses it to characterize range-kernel complementarity, providing new geometric insights into operator theory.

## Key findings

- Range-kernel complementarity characterized by the angle being less than π.
- Operator range is closed when the angle condition is satisfied.
- The angle concept applies when 0 faces the unbounded component of the resolvent set.

## Abstract

In this paper, we define the angle of a bounded linear operator $A$ along an unbounded path emanating from the origin and use it to characterize range-kernel complementarity. In particular we show that if $0$ faces the unbounded component of the resolvent set, then $X=R(A)\oplus N(A)$ if and only if $R(A)$ is closed and some angle of $A$ is less than $\pi$.