Range-Kernel Complementation

TL;DR

This paper explores the relationships between complemented ranges and kernels of Banach-space operators and their adjoints, providing conditions and applications to compact operators and perturbations.

Contribution

It establishes new equivalences between complemented ranges and kernels for operators and their adjoints, especially in the context of compact perturbations and semi-Fredholm operators.

Findings

Complemented range implies complemented kernel for the adjoint in reflexive Banach spaces.

Complemented kernel for an operator is equivalent to complemented range for its adjoint under certain conditions.

Compact perturbations of semi-Fredholm operators have complemented range and kernel for both the operator and its adjoint.

Abstract

If a Banach-space operator has a complemented range, then its normed-space adjoint has a complemented kernel and the converse holds on a reflexive Banach space. It is also shown when complemented kernel for an operator is equivalent to complemented range for its normed-space adjoint. This is applied to compact operators and to compact perturbations. In particular, compact perturbations of semi-Fredholm operators have complemented range and kernel for both the perturbed operator and its normed-space adjoint