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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A note on commutativity up to a factor of bounded operators


Authors: Jian Yang and Hong-Ke Du
Translated by:
Journal: Proc. Amer. Math. Soc. 132 (2004), 1713-1720
MSC (2000): Primary 47A10
Published electronically: January 7, 2004
MathSciNet review: 2051132
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Abstract: In this note, we explore commutativity up to a factor $AB=\lambda BA$ for bounded operators $A$ and $B$ in a complex Hilbert space. Conditions on possible values of the factor $\lambda$ are formulated and shown to depend on spectral properties of the operators. Commutativity up to a unitary factor is considered. In some cases, we obtain some properties of the solution space of the operator equation $AX=\lambda XA$ and explore the structures of $A$ and $B$ that satisfy $AB=\lambda BA$ for some $\lambda \in \mathbb{C}\setminus \{ 0 \}.$ A quantum effect is an operator $A$on a complex Hilbert space that satisfies $0\leq A \leq I.$ The sequential product of quantum effects $A$ and $B$ is defined by $A\circ B=A^{\frac{1}{2}}BA^{\frac{1}{2}}.$ We also obtain properties of the sequential product.


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Additional Information

Jian Yang
Affiliation: College of Mathematics and Information Science, Shaanxi Normal University, Xi’an 710062, P. R. China
Email: yangjia0426@sina.com

Hong-Ke Du
Affiliation: College of Mathematics and Information Science, Shaanxi Normal University, Xi’an 710062, P. R. China
Email: hkdu@snnu.edu.cn

DOI: http://dx.doi.org/10.1090/S0002-9939-04-07224-7
PII: S 0002-9939(04)07224-7
Keywords: Hilbert space, commutator, selfadjointness, normal operator, quantum effect
Received by editor(s): October 25, 2002
Received by editor(s) in revised form: January 9, 2003
Published electronically: January 7, 2004
Additional Notes: This work was partially supported by the National Natural Science Foundation of China
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2004 American Mathematical Society