  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.
