zbMATH — the first resource for mathematics

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)].)

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.)
Full Text: DOI Link