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The supremum of linear operators for the *-order. (English) Zbl 1204.47022

Let \(\mathcal{B(H)}\) be the algebra of all bounded linear operators on a complex Hilbert space \(\mathcal{H}\). For \(A,B\in \mathcal{B(H)}\), the \(*\)-order \(A\leq^{*} B\) is defined by \(A^{*}A=A^{*}B\) and \(AA^{*}=BA^{*}\). In this paper, the authors obtain a characterization of operators \(C\in \mathcal{B(H)}\) such that \(A\leq^{*} C\) and \(B\leq^{*} C\) (if such \(C\) exists). Moreover, the authors obtain the exact form of the least upper bound of \(A\) and \(B\) with respect to the \(*\)-order, i.e.,
\[ A\bigvee^{*} B=\min \{C\in \mathcal{B(H)}; A\leq^{*} C \text{ and }B\leq^{*} C\}, \]
where \(\min\) denotes the minimum operator respect to the \(*\)-order, and obtain a necessary and sufficient condition under which \(A\overset {*}\bigvee B\) exists.

MSC:

47A63 Linear operator inequalities
47N50 Applications of operator theory in the physical sciences
81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations
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References:

[1] J. Antezana, C. Cano, I. Mosconi, D. Stojanoff, A note on the star order in Hilbert spaces, Linear and Multilinear Algebra, in press, doi:10.1080/03081080903227104; J. Antezana, C. Cano, I. Mosconi, D. Stojanoff, A note on the star order in Hilbert spaces, Linear and Multilinear Algebra, in press, doi:10.1080/03081080903227104 · Zbl 1203.47010
[2] Baksalary, J. K.; Baksalary, O. M.; Liu, X. J., Further properties of the star, left-star, right-star, and minus partial orderings, Linear Algebra Appl., 375, 83-94 (2003) · Zbl 1048.15016
[3] Conway, J. B., A Course in Functional Analysis (1989), Springer-Verlag: Springer-Verlag New York
[4] Deng, C. Y.; Du, H. K., Common complements of two subspaces and an answer to Groß’s question, Acta Math. Sin. (Chinese Ser.), 49, 1099-1112 (2006) · Zbl 1211.47002
[5] Drazin, M. P., Natural structures on semigroups with involution, Bull. Amer. Math. Soc., 84, 139-141 (1978) · Zbl 0395.20044
[6] Gudder, S., An order for quantum observables, Math. Slovaca, 56, 573-589 (2006) · Zbl 1141.81008
[7] Halmos, P., Two subspaces, Trans. Amer. Math. Soc., 144, 381-389 (1969) · Zbl 0187.05503
[8] Hartwig, R. E.; Drazin, M. P., Lattice properties of the ∗-order for complex matrices, J. Math. Anal. Appl., 86, 359-378 (1982) · Zbl 0485.15014
[9] Liu, W. H.; Wu, J. D., A representation theorem of infimum of bounded quantum observables, J. Math. Phys., 49, 073521 (2008) · Zbl 1152.81631
[10] Pulmannová, S.; Vincenková, E., Remarks on the order for quantum observables, Math. Slovaca, 57, 589-600 (2007) · Zbl 1164.81001
[11] Xu, X. M.; Du, H. K.; Fang, X. C., An explicit expression of supremum of bounded quantum observables, J. Math. Phys., 50, 033502 (2009) · Zbl 1202.81014
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