Title:
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Dilations of positive operator measures and bimeasures related to quantum mechanics (English) |
Author:
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Lahti, Pekka |
Author:
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Ylinen, Kari |
Language:
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English |
Journal:
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Mathematica Slovaca |
ISSN:
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0139-9918 |
Volume:
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54 |
Issue:
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2 |
Year:
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2004 |
Pages:
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[169]-189 |
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Category:
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math |
. |
MSC:
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54F05 |
MSC:
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81P15 |
idZBL:
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Zbl 1071.81014 |
idMR:
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MR2074214 |
. |
Date available:
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2009-09-25T14:19:39Z |
Last updated:
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2012-08-01 |
Stable URL:
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http://hdl.handle.net/10338.dmlcz/136901 |
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Reference:
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[1] ANDERSEN E. S.-JESSEN B.: On the introduction of measures in infinite product sets.Danske Vid. Selsk. Mat.-Fys. Medd. 25 (1948). Zbl 0031.01305, MR 0027041 |
Reference:
|
[2] BERBERIAN S. K.: Notes on Spectral Theory.D. van Nostrand Co., Inc, Princeton, 1966. Zbl 0138.39104, MR 0190760 |
Reference:
|
[3] BERG C.-CHRISTENSEN J. P. R.-RESSEL P.: Harmonic Analysis on Semigroups.Grad. Texts in Math. 100, Springer, New York, 1984. Zbl 0619.43001, MR 0747302 |
Reference:
|
[4] BIRMAN M.-SOLOMYAK M.: Tensor products of a finite number of spectral measures is always a spectral measure.Integral Equations Operator Theory 24 (1996), 179-187. MR 1371945 |
Reference:
|
[5] BIRMAN M. SH.-VERSHIK A. M.-SOLOMYAK M. Z.: Product of commuting spectral measures need not be countably additive.Funktsional Anal. i Prilozhen. 13(1) (1978), 61-66 [English translation: Functional Anal. Appl. 13 (1979), 48-49]. MR 0527523 |
Reference:
|
[6] CASSINELLI G.-DE VITO E.: Square-integrability modulo a subgroup.Trans. Amer. Math. Soc. 355 (2003), 1443-1465. MR 1946399 |
Reference:
|
[7] CASSINELLI G.-DE VITO E.-LAHTI P.-LEVRERO A.: Phase space observables and isotypic spaces.J. Math. Phys. 41 (2000), 5883-5896. MR 1779140 |
Reference:
|
[8] CASSINELLI G.-DE VITO E.-LAHTI P.-PELLONPÄÄ J.-P.: Covariant localizations in the torus and the phase observables.J. Math. Phys. 43 (2002), 693-704. MR 1878565 |
Reference:
|
[9] CHEN P. D.-LI J. F.: On the existence of product stochastic measures.Acta Math. Appl. Sinica 7 (1991), 120-134. Zbl 0735.60001, MR 1105385 |
Reference:
|
[10] DAVIES E. B.: The Quantum Theory of Open Systems.Academic Press, London-New York-San Francisco, 1976. MR 0489429 |
Reference:
|
[11] DAVIES E. B.-LEWIS J. T.: An operational approach to quantum probability.Comm. Math. Phys. 17 (1970), 239-260. Zbl 0194.58304, MR 0263379 |
Reference:
|
[12] DOUGLAS R. G.: Banach Algebra Techniques in Operator Theory.Academic Press, New York, 1972. Zbl 0247.47001, MR 0361893 |
Reference:
|
[13] DUDLEY R. M.: A note on products of spectral measures.In: Vector and Operator Valued Measures and Applications (Proc. Sympos., Alta, Utah, 1972), Academic Press, New York, 1973, pp. 125-126. MR 0336418 |
Reference:
|
[14] DUNFORD N.-SCHWARTZ J. T.: Linear Operators, Part I: General Theory.Interscience Publishers, New York, 1958. MR 1009162 |
Reference:
|
[15] DVURECENSKIJ A.-LAHTI P.-YLINEN K.: Positive operator measures determined by their moment operators.Rep. Math. Phys. 45 (2000), 139-146. MR 1751508 |
Reference:
|
[16] GARRISON J. C.-WONG J.: Canonically conjugate pairs, uncertainty relations, and phase operators.J. Math. Phys. 11 (1970), 2242-2249. Zbl 0196.28003, MR 0266532 |
Reference:
|
[17] HALMOS P. R.: Measure Theory.Van Nostrand, Toronto, 1950. Zbl 0040.16802, MR 0033869 |
Reference:
|
[18] HOLEVO A. S.: Probabilistic and Statistical Aspects of Quantum Theory.North Holland, Amsterdam, 1982. Zbl 0497.46053, MR 0681693 |
Reference:
|
[19] KARNI S.-MERZBACH E.: On the extension of bimeasures.J. Anal. Math. 55 (1990), 1-16. Zbl 0724.28003, MR 1094708 |
Reference:
|
[20] KRAUS K.: States, Effects, and Operations.Springer-Verlag, Berlin, 1983. Zbl 0545.46049, MR 0725167 |
Reference:
|
[21] LAHTI P.-MACZYŃSKI M.-YLINEN K.: The moment operators of phase space observables and their number margins.Rep. Math. Phys. 41 (1998), 319-331. MR 1653885 |
Reference:
|
[22] LAHTI P.-PELLONPAA J.-P.-YLINEN K.: Operator integrals and phase space observables.J. Math. Phys. 40 (1999), 2181-2189. Zbl 0984.47054, MR 1683153 |
Reference:
|
[23] LAHTI P.-PULMANNOVA S.: Coexistent observables and effects in quantum mechanics.Rep. Math. Phys. 39 (1997), 339-351. Zbl 0924.47054, MR 1477898 |
Reference:
|
[24] LAHTI P.-PULMANNOVA S.: Coexistence vs. functional coexistence of quantum observables.Rep. Math. Phys. 47 (2001), 199-212. Zbl 1005.81006, MR 1836331 |
Reference:
|
[25] LAHTI P.-PULMANNOVA S.-YLINEN K.: Coexistent observables and effects in a convexity approach.J. Math. Phys. 39 (1998), 6364-6371. Zbl 0935.81010, MR 1656976 |
Reference:
|
[26] LERNER E. C.-HUANG H. W.-WALTERS G. E.: Some mathematical properties of oscillator phase operators.J. Math. Phys. 11 (1970), 1679-1684. MR 0260346 |
Reference:
|
[27] LEWIS D. R.: Integration with respect to vector measures.Pacific J. Math. 33 (1970), 157-165. Zbl 0195.14303, MR 0259064 |
Reference:
|
[28] LUDWIG G.: Foundations of Quantum Mechanics.Vol. I, Springer-Verlag, Berlin, 1983. Zbl 0509.46057, MR 0690770 |
Reference:
|
[29] MLAK W.: Notes on quantum circular operators.Preprint 303, Institute of Mathematics, Polish Academy of Sciences, 1984. |
Reference:
|
[30] MARCZEWSKI E.-RYLL-NARDZEWSKI C.: Remarks on the compactness and non direct products of measures.Fund. Math. 40 (1953), 165-170. Zbl 0052.05001, MR 0059996 |
Reference:
|
[31] NAIMARK M. A.: On a representation of additive operator set functions.Comptes Rendus (Doklady) Acad. Sci. URSS 41 (1943), 359-361. Zbl 0061.25410, MR 0010789 |
Reference:
|
[32] OZAWA M.: Quantum measuring processes of continuous observables.J. Math. Phys. 25 (1984), 79-87. MR 0728889 |
Reference:
|
[33] OZAWA M.: Phase operator problem and macroscopic extension of quantum mechanics.Ann. Physics 257 (1997), 65-83. Zbl 0921.03059, MR 1455986 |
Reference:
|
[34] PAULSEN V. I.: Completely Bounded Maps and Dilations.Pitman Res. Notes Math. Ser. 146, Longman, London, 1986. Zbl 0614.47006, MR 0868472 |
Reference:
|
[35] PELLONPÄÄ J.-P.: Covariant phase observables in quantum mechanics.Ann. Univ. Turku. Ser. A I no. 288 (2002). Zbl 0969.81003, MR 1975169 |
Reference:
|
[36] PUTNAM C. P. : Commutation Properties of Hilbert Space Operators and Related Topics.Springer-Verlag, Berlin, 1967. MR 0217618 |
Reference:
|
[37] RIESZ F.-SZ.-NAGY B.: Functional Analysis.Dover Publications, Inc., New York, 1990. Zbl 0732.47001, MR 1068530 |
Reference:
|
[38] RUDIN W.: Functional Analysis.McGraw Hill, New York, 1973. Zbl 0253.46001, MR 0365062 |
Reference:
|
[39] STINESPRING W. F.: Positive functions on $C^\ast$-algebras.Proc. Amer. Math. Soc. 6 (1955), 211-216. Zbl 0064.36703, MR 0069403 |
Reference:
|
[40] STULPE W.: Classical Representations of Quantum Mechanics Related to Statistically Complete Observables.Wissenschaft und Technik Verlag, Berlin, 1997. Zbl 0898.46069 |
Reference:
|
[41] VARADARAJAN V. S.: Geometry of Quantum Theory.Springer, Berlin, 1985. Zbl 0581.46061, MR 0805158 |
Reference:
|
[42] YLINEN K.: On vector bimeasures.Ann. Mat. Pura Appl. 117 (1978), 115-138. MR 0515957 |
Reference:
|
[43] YLINEN K.: Positive operator bimeasures and a noncommutative generalization.Studia Math. 118 (1996), 157-168. Zbl 0399.46032, MR 1389762 |
Reference:
|
[44] YLINEN K.: Aspects of operator measures and bimeasures susceptible to a nonstandard treatment.In: Mathematisches Forschungsinstitut Oberwolfach, Tagungsbericht 08/1999, Nonstandard Analysis and Related Methods, and their Applications 21. 2.-27. 2. 1999. http://www.mfo.de/Meetings/Documents/1999/08/ReportJ08_99.ps. Zbl 0862.46038 |
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