A note on commutativity up to a factor of bounded operators
Authors:
Jian Yang and HongKe Du
Translated by:
Journal:
Proc. Amer. Math. Soc. 132 (2004), 17131720
MSC (2000):
Primary 47A10
Published electronically:
January 7, 2004
MathSciNet review:
2051132
Fulltext PDF Free Access
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Abstract: In this note, we explore commutativity up to a factor for bounded operators and in a complex Hilbert space. Conditions on possible values of the factor 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 and explore the structures of and that satisfy for some A quantum effect is an operator on a complex Hilbert space that satisfies The sequential product of quantum effects and is defined by We also obtain properties of the sequential product.
 1.
James
A. Brooke, Paul
Busch, and David
B. Pearson, Commutativity up to a factor of bounded operators in
complex Hilbert space, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng.
Sci. 458 (2002), no. 2017, 109–118. MR 1879460
(2003b:47034), http://dx.doi.org/10.1098/rspa.2001.0858
 2.
Paul
Busch, Unsharp localization and causality in relativistic quantum
theory, J. Phys. A 32 (1999), no. 37,
6535–6546. MR 1733620
(2001a:81093), http://dx.doi.org/10.1088/03054470/32/37/305
 3.
Paul
Busch, Pekka
J. Lahti, and Peter
Mittelstaedt, The quantum theory of measurement, Lecture Notes
in Physics. New Series m: Monographs, vol. 2, SpringerVerlag, Berlin,
1991. MR
1176754 (93m:81014)
 4.
P. Busch and J. Singh, Lüder theorem for unsharp quantum measurements, Phys. Lett., A249 (1998), 1012.
 5.
John
B. Conway, A course in functional analysis, Graduate Texts in
Mathematics, vol. 96, SpringerVerlag, New York, 1985. MR 768926
(86h:46001)
 6.
E.
B. Davies, Quantum theory of open systems, Academic Press
[Harcourt Brace Jovanovich, Publishers], LondonNew York, 1976. MR 0489429
(58 #8853)
 7.
A. Gheondea and S. Gudder, Sequential product of quantum effects, Proc. Amer. Math. Soc., to appear.
 8.
Stan
Gudder and Gabriel
Nagy, Sequentially independent
effects, Proc. Amer. Math. Soc.
130 (2002), no. 4,
1125–1130. MR 1873787
(2002i:81014), http://dx.doi.org/10.1090/S0002993901061949
 9.
Stan
Gudder and Gabriel
Nagy, Sequential quantum measurements, J. Math. Phys.
42 (2001), no. 11, 5212–5222. MR 1861337
(2002h:81032), http://dx.doi.org/10.1063/1.1407837
 10.
Roger
A. Horn and Charles
R. Johnson, Matrix analysis, Cambridge University Press,
Cambridge, 1985. MR 832183
(87e:15001)
 11.
C.
R. Putnam, Commutation properties of Hilbert space operators and
related topics, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band
36, SpringerVerlag New York, Inc., New York, 1967. MR 0217618
(36 #707)
 12.
John
von Neumann, Mathematical foundations of quantum mechanics,
Princeton University Press, Princeton, 1955. Translated by Robert T. Beyer.
MR
0066944 (16,654a)
 1.
 J. A. Brooke, P. Busch and B. Pearson, Commutativity up to a factor of bounded operators in complex Hilbert space, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., A458 (2002), no. 2017, 109118. MR 2003b:47034
 2.
 P. Busch, Unsharp localization and causality in relativistic quantum theory, J. Phys., A32 (1999), no. 37, 65356546. MR 2001a:81093
 3.
 P. Busch, P. J. Lahti and P. Mittlestaedt, The quantum theory of measurement, SpringerVerlag, Berlin, 1991. MR 93m:81014
 4.
 P. Busch and J. Singh, Lüder theorem for unsharp quantum measurements, Phys. Lett., A249 (1998), 1012.
 5.
 J. B. Conway, A course in functional analysis, Graduate Texts in Mathematics, No. 96, SpringerVerlag, New York, 1985. MR 86h:46001
 6.
 E. B. Davies, Quantum theory of open systems, Academic Press, LondonNew York, 1976. MR 58:8853
 7.
 A. Gheondea and S. Gudder, Sequential product of quantum effects, Proc. Amer. Math. Soc., to appear.
 8.
 S. Gudder and G. Nagy, Sequentially independent effects, Proc. Amer. Math. Soc., 130 (2002), no. 4, 11251130. MR 2002i:81014
 9.
 S. Gudder and G. Nagy, Sequential quantum measurements, J. Math. Phys., 42 (2001), 52125222. MR 2002h:81032
 10.
 R. A. Horn and C. R. Johnson, Matrix analysis, Cambridge Univ. Press, Cambridge, 1985. MR 87e:15001
 11.
 C. R. Putnam, Commutation properties of Hilbert space operators and related topics, SpringerVerlag, New York, 1967. MR 36:707
 12.
 J. von Neumann, Mathematical foundations of quantum mechanics, Princeton Univ. Press, Princeton, New Jersey, 1955. MR 16:654a
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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
HongKe 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/S0002993904072247
PII:
S 00029939(04)072247
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
