On an unconditional basis with parentheses for generalized subordinate perturbations and application to Gribov operators in Bargmann space

In this article, we give a new definition which generalizes the notion of subordination between two operators. Moreover, we give a description of the changed spectrum and we establish different conditions in terms of the spectrum to prove the existence of unconditional basis with parentheses. We apply this work to some block operators matrix. The obtained results are of importance to be applied to Gribov operators in Bargmann space.