Complex-self-adjointness

TL;DR

This paper introduces a new class of operators in Hilbert spaces that are similar to their adjoints through antiunitary operators, expanding the mathematical framework for non-self-adjoint operators relevant in quantum physics.

Contribution

It develops the theory of complex-self-adjoint operators, including extension theory, decompositions, and eigenfunction expansions, motivated by quantum mechanical symmetries.

Findings

Extended the concept of self-adjointness via antiunitary operators

Provided new decomposition methods for these operators

Linked mathematical theory to physical symmetries in quantum mechanics

Abstract

We develop the concept of operators in Hilbert spaces which are similar to their adjoints via antiunitary operators, the latter being not necessarily involutive. We discuss extension theory, refined polar and singular-value decompositions, and antilinear eigenfunction expansions. The study is motivated by physical symmetries in quantum mechanics with non-self-adjoint operators.