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Maps on operators strongly preserving sharp order. (English) Zbl 1345.15007
Let $${\mathscr B}(H)$$ be the algebra of bounded linear operators on a real or complex Hilbert space $$H$$ with $$\dim H\geq 3$$.
P. Šemrl [J. Math. Anal. Appl. 369, No. 1, 205–213 (2010; Zbl 1195.47027)] showed that skew-projections can be used to extend the minus partial order from matrices to $${\mathscr B}(H)$$. Following the same idea, M. A. Efimov [Math. Notes 93, No. 5, 784–788 (2013); translation from Mat. Zametki 93, No. 5, 794–797 (2013; Zbl 1303.47003)] extended the sharp partial order from matrices to $${\mathscr B}(H)$$ by $$A\leq^\sharp B$$ if $$A=B$$ or if there exists a skew-projection $$P\in {\mathscr B}(H)$$ such that $\text{Im}\,P=\overline{\text{Im}\,A},\quad\text{Ker} P=\text{Ker} A, \quad PA=PB,\quad\text{and}\quad AP=BP.$ Presently, the authors classify additive bijections on $${\mathscr B}(H)$$ that are strongly monotone with respect to $$\leq^\sharp$$ partial order. It is shown that such maps take the form $$X\mapsto \alpha SXS^{-1}$$ or $$X\mapsto \alpha SX^\ast S^{-1}$$ for some semilinear bijection $$S: H\to H$$ (which is linear or conjugate-linear and bounded if $$\dim H=\infty$$) and some nonzero scalar $$\alpha$$. A similar result, but without imposing additivity, is derived for strongly $$\leq^\sharp$$ monotone bijections from the set of linear spans of pairwise orthogonal skew-projections onto itself.
If the bijectivity or additivity assumptions are dropped, then the structure of strongly $$\leq^\sharp$$ monotone maps on $${\mathscr B}(H)$$ is more complicated. The authors show, for example, that a linear map $$X\mapsto B^\ast XB$$, where $$B$$ is a backwards shift on $$H=\ell^2$$, is strongly $$\leq^\sharp$$ monotone. An example of a nonadditive bijective strongly $$\leq^\sharp$$ monotone map on $${\mathscr B}(H)$$ is also given.
(Reviewer’s remark: It was recently established by G. Dolinar, the reviewer, and J. Marovt [“A note on partial orders of Hartwig, Mitsch, and Šemr”, Appl. Math. Comput. 270, 711–713 (2015; doi:10.1016/j.amc.2015.08.066).] that Šemrl’s extension of minus partial order coincides with a natural partial order defined on general semigroups by H. Mitsch [Proc. Am. Math. Soc. 97, 384–388 (1986; Zbl 0596.06015)].)

##### MSC:
 15A86 Linear preserver problems 06F05 Ordered semigroups and monoids 15B48 Positive matrices and their generalizations; cones of matrices 15A04 Linear transformations, semilinear transformations 47A05 General (adjoints, conjugates, products, inverses, domains, ranges, etc.)
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