  We consider similarity transformations of a perturbed linear operator $A-B$
in a complex Banach space $\mathcal{X}$, where the unperturbed operator $A$ is
a generator of a Banach $L_1(\mathbb{R})$-module and the perturbation operator
$B$ is a bounded linear operator. The result of the transformation is a simpler
operator $A-B_0$. For example, if $A$ is a differentiation operator and $B$ is
an operator of multiplication by an operator-valued function, then $B_0$ is an
operator of multiplication by a function that is a restriction of an entire
function of exponential type and could be $0$ in some cases. As a consequence,
we derive the spectral invariance of the operator $A-B$ in a large class of
spaces. The study is based on a widely applicable modification of the method of
similar operators that is also presented in the paper. This non-traditional
modification is rooted in the spectral theory of Banach
$L_1(\mathbb{R})$-modules.
