Locally bounded perturbations and (odd) unbounded KK-theory

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.