Kernels of paired operators and their adjoints

TL;DR

This paper reviews the properties of paired operators and their adjoints, focusing on kernels, commutation relations, and invariance properties, highlighting both similarities and differences.

Contribution

It extends known invariance results to paired operators and provides a detailed comparison of kernels and adjoints, emphasizing their unique features.

Findings

Kernels of paired operators and their adjoints have both similarities and stark differences.

Invariance properties can be extended to paired operators.

The study clarifies the relationship between paired operators and their transposes.

Abstract

We review the basic properties of paired operators and their adjoints, the transposed paired operators, with particular reference to commutation relations, and we study the properties of their kernels, bringing out their similarities and also, somewhat surprisingly, their stark differences. Various notions expressing different invariance properties are also reviewed and we extend to paired operators some known invariance results.