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We study the star and minus partial orders on the set $ {\mathcal {B}}(H,K) $ B (H , K) of all bounded operators acting from a Hilbert space H to a Hilbert space K. We extend and strengthen some results from matrix case to the case of general operators, which may not possess generalized inverses. By means of some norm inequalities, we give necessary and sufficient condition under which two operators have orthogonal ranges, and thus, we give a characterization of the star partial order. When A = PB for some projection P, we prove the equivalence of $ A \lt ^*B $ A < ∗ B with $ f(AA^*)A \lt ^*f(BB^*)B $ f (A A ∗) A < ∗ f (B B ∗) B for a wide class of continuous functions f. Also, we prove that $ A \lt ^*B $ A < ∗ B if and only if $ A \lt ^-B $ A < − B and $ A^2 \lt ^{-}B^{2} $ A 2 < − B 2 when A is a weak EP operator and B is a self-adjoint operator. Finally, we consider the Moore–Penrose invertibility and the ordinary invertibility of a linear combination of operators when they are related with one of these two orders. [ABSTRACT FROM AUTHOR]

The unified theory for matrix partial orders based on generalized inverses has already been done by Mitra. We consider a special kind of ring R , which generalizes the ring of linear operators on finite dimensional vector space, and extend Mitra's approach to it. Thus, we find necessary and sufficient condition for a G -based relation to be a partial order on R . Also, some known results are generalized and some new results are proved. [ABSTRACT FROM AUTHOR]

Let R be a ring with involution. The recently introduced notions of the core and dual core inverse are extended from matrix to an arbitrary ⁎-ring case. It is shown that the group, Moore–Penrose, core and dual core inverse are closely related and they can be treated in the same manner using appropriate idempotents. The several characterizations of these inverses are given. Some new properties are obtained and some known results are generalized. A number of characterizations of EP elements in R are obtained. It is shown that core and dual core inverse belong to the class of inverses along an element and to the class of ( b , c ) -inverses. [ABSTRACT FROM AUTHOR]

The core inverse of matrix is generalized inverse which is in some sense in-between the group and Moore–Penrose inverse. In this paper a generalization of core inverse and core partial order to Hilbert space operator case is presented. Some properties are generalized and some new ones are proved. Connections with other generalized inverses are obtained. The useful matrix representations of operator and its core inverse are given. It is shown that A is less than B under the core partial order if and only if they have specific kind of simultaneous diagonalization induced by appropriate decompositions of Hilbert space. The relation is also characterized by the inclusion of appropriate sets of generalized inverses. The spectral properties of core inverse are also obtained. [ABSTRACT FROM AUTHOR]