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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 aim of this short note is to indicate that Rao and Mitra's constrained matrix inverse A c r C R (introduced in 1972) is a direct precursor of recently introduced Drazin's ( b , c ) outer generalized inverse. Precisely, if one rewrites definition of A c r C R in purely multiplicative language, then it turns out that two inverses coincide. Recently introduced Mary's inverse along an element and Manjunatha Prasad and Mohana's core-EP inverse are the special cases of Rao and Mitra's constrained inverse. [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]