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A note on majorization and range inclusion of adjointable operators on Hilbert \(C^\ast\)-modules. (English) Zbl 1364.46053

X.-C. Fang et al. [Linear Algebra Appl. 431, No. 11, 2142–2153 (2009; Zbl 1175.47014)] extended the celebrated majorization theorem of R. G. Douglas [Proc. Am. Math. Soc. 17, 413–415 (1966; Zbl 0146.12503)] as follows: Let \(C\in \mathcal{L}(\mathscr{G},\mathscr{F})\), \(A\in \mathcal{L}(\mathscr{E},\mathscr{F})\) and \(\overline{\mathcal{R}(A^*)}\) be orthogonally complemented. Then the following statements are equivalent:
(1)
\(CC^*\leq \lambda AA^*\) for some \(\lambda > 0.\)
(2)
There exists \(\mu > 0\) such that \(\| C^*z\|\leq \mu \| A^*z\|\) for any \(z\in \mathscr{F}\).
(3)
There exists \(D\in \mathcal{L}(\mathscr{G},\mathscr{E})\) such that \(AD=C\), i.e., \(AX=C\) has a solution.
(4)
\(\mathcal{R}(C)\subseteq \mathcal{R}(A)\).
In this paper, the authors give a nice counterexample to show that the conditions (1) and (4) are not equivalent. For another counterexample showing that (2) and (4) are not equivalent and some related discussion, see also [Z. Mousavi et al., Linear Algebra Appl. 517, 85–98 (2017; Zbl 1355.15013)].

MSC:

46L08 \(C^*\)-modules
47A05 General (adjoints, conjugates, products, inverses, domains, ranges, etc.)
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References:

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