Method of similar operators in the problem of bi-invariant subspaces

Abstract

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.