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

In this paper, we discuss the construction of bi-invariant subspaces for a self-adjoint, linear, closed operator with discrete spectrum perturbed by a bounded operator. The main result is the theorem on the similarity of this operator to a block diagonal operator. This theorem implies results concerning biinvariant subspaces and formulas for projectors and weighted average eigenvalues. In addition, we construct the corresponding group of operators and propose a new modification of the method of similar operators.