A Unified Approach to the Decomposition Theorems in Baer $$*$$-Rings

Abstract

AbstractThe aim of the paper is to generalize decomposition theorems showed in Bagheri-Bardi et al. (Linear Algebra Appl 583:102–118, 2019; Linear Algebra Appl 539:117–133, 2018) by a unified approach. We show a general decomposition theorem with respect to a hereditary property. Then the vast majority of decompositions known in the algebra of Hilbert space operators is generalized to elements of Baer $$*$$ ∗ -rings by this theorem. The theorem yields also results which are new in the algebra of bounded Hilbert space operators. Additionally, the model of summands in Wold–Słociński decomposition is given in Baer $$*$$ ∗ -rings.