arXiv subjects

Martin Z. Ljubenović

Publications and source records attributed to Martin Z. Ljubenović.

2 recordsLinked to original sources

arXiv2022

1MP and MP1 inverses and one-sided star orders in a ring with involution

The classes of 1MP-inverses and MP1-inverses are recently introduced classes of generalized inverses of complex matrix. Actually, they coincide with the classes of $\{1,2,3\}$ and $\{1,2,4\}$ inverses, respectively. We consider these inverses in the context of a ring with involution and prove that their most important characterizations and properties remain true. We show that the binary relations based on these inverses are in fact the well known left-star and right-star partial orders. We extend these relations to the ring case, connect them with the unified theory of partial order relations based on generalized inverses and provide several properties. Finally, we indicate how these results can be applied to bounded Hilbert space operators.

math.RA↗

arXiv2021

Submajorization on $\ell^p(I)^+$ determined by increasable doubly substochastic operators and its linear preservers

We note that the well-known result of Von Neumann \cite{von} is not valid for all doubly substochastic operators on discrete Lebesgue spaces $\ell^p(I)$, $p\in[1,\infty)$. This fact lead us to distinguish two classes of these operators. Precisely, the class of increasable doubly substochastic operators on $\ell^p(I)$ is isolated with the property that an analogue of the Von Neumann result on operators in this class is true. The submajorization relation $\prec_s$ on the positive cone $\ell^p(I)^+$, when $p\in[1,\infty)$, is introduced by increasable substochastic operator and it is provided that submajorization may be considered as a partial order. Two different shapes of linear preservers of submajorization $\prec_s$ on $\ell^1(I)^+$ and on $\ell^p(I)^+$, when $I$ is an infinite set, are presented.

math.FA↗