Method of similar operators in harmonious Banach spaces

We consider similarity transformations of a perturbed linear operator $A-B$ in a complex Banach space X, where the unperturbed operator A is a generator of a Banach L₁(R)-module and the perturbation operator B is a bounded linear operator. The result of the transformation is a simpler operator A-B₀. For example, if A is a differentiation operator and B is an operator of multiplication by an operator-valued function, then B₀ 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 L₁(R)-modules.