  The theory of quaternionic operators has applications in several different
fields such as quantum mechanics, fractional evolution problems, and
quaternionic Schur analysis, just to name a few. The main difference between
complex and quaternionic operator theory is based on the definition of
spectrum. In fact, in quaternionic operator theory the classical notion of
resolvent operator and the one of spectrum need to be replaced by the two
$S$-resolvent operators and the $S$-spectrum. This is a consequence of the
non-commutativity of the quaternionic setting. Indeed, the $S$-spectrum of a
quaternionic linear operator $T$ is given by the non invertibility of a second
order operator. This presents new challenges which makes our approach to
perturbation theory of quaternionic operators different from the classical
case. In this paper we study the problem of perturbation of a quaternionic
normal operator in a Hilbert space by making use of the concepts of
$S$-spectrum and of slice hyperholomorphicity of the $S$-resolvent operators.
For this new setting we prove results on the perturbation of quaternionic
normal operators by operators belonging to a Schatten class and give conditions
which guarantee the existence of a nontrivial hyperinvariant subspace of a
quaternionic linear operator.

Functional Analysis