Partial orders based on generalized inverses and annihilators

The dissertation is a contribution in defining and studying different types of partial orders on rings, particularly on von Neumann regular and Rickart rings, as well as on the algebras of bounded operators on Banach and Hilbert spaces. In the first part of the dissertation, using generalized inverses of elements, the definitions of the minus, star, sharp, core and dual core partial orders are extended from the set of complex matrices to an arbitrary ring (with or without an involution). In a unified approach it is shown that the condition a<b, where < is one of the above mentioned relation, or any other G relation, defines two decompositions of the identity of the ring, and thus, the representations of a and b in the diagonal 3×3 matrix forms. It is shown that the most important properties of the matrix partial orders based on generalized inverses stay valid in the ring case. Moreover, the considerable number of new results are proved. In the second part of the dissertation, the above mentioned relations are characterized using the notion of annihilators. This approach allows us to define the relations for elements which do not possess a generalized inverse. Partial orders based on annihilators are specially investigated on the Rickart and Rickart *-rings. Both approaches give as a corollary the appropriate results on the algebras of bounded operators on Banach or Hilbert spaces. The main ideas in the dissertation are precisely motivated by this case