Locally bounded perturbations and (odd) unbounded KK-theory

TL;DR

This paper establishes that locally bounded perturbations preserve the regularity and self-adjointness of unbounded operators on Hilbert modules, ensuring stability of Kasparov classes and enabling new module constructions.

Contribution

It proves that locally bounded symmetric perturbations maintain regularity and self-adjointness, and applies this to stability of Kasparov classes and module doubling procedures.

Findings

Locally bounded perturbations preserve regularity and self-adjointness.

Kasparov classes are stable under locally bounded perturbations.

Constructed unbounded multipliers yield operators with compact resolvent after doubling.

Abstract

A regular symmetric operator on a Hilbert module is self-adjoint whenever there exists a suitable approximate identity. We say an operator is 'locally bounded' if the composition of the operator with each element in the approximate identity is bounded. We prove that the perturbation of a regular self-adjoint operator by a locally bounded symmetric operator is again regular and self-adjoint. We use this result to show that the Kasparov class represented by an unbounded Kasparov module is stable under locally bounded perturbations. As an application, we show that we obtain a converse to the 'doubling up' procedure of odd unbounded Kasparov modules. Finally, we discuss perturbations of unbounded Kasparov modules by unbounded multipliers. In particular, we explicitly construct an unbounded multiplier such that (after doubling up the module) the perturbed operator has compact resolvent.