A note on commutativity up to a factor of bounded operators
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- by Jian Yang and Hong-Ke Du
- Proc. Amer. Math. Soc. 132 (2004), 1713-1720
- DOI: https://doi.org/10.1090/S0002-9939-04-07224-7
- Published electronically: January 7, 2004
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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.References
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Bibliographic 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
- 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
- © Copyright 2004 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 132 (2004), 1713-1720
- MSC (2000): Primary 47A10
- DOI: https://doi.org/10.1090/S0002-9939-04-07224-7
- MathSciNet review: 2051132