Abstract
Some new sufficient conditions are found on operators A, B in a Hilbert space, under which trace (AB−BA)=0.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Literature cited
M. Sh. Birman and M. Z. Solomyak, “Estimates of singular numbers of integral operators,” Usp. Mat. Nauk,32, No. 1, 17–84 (1977).
B. Fuglede, “A commutativity theorem for normal operators,” Proc. Nat. Acad. Sci. USA,36, 35–40 (1950).
P. Halmos, “Ten problems in Hilbert spaces,” Bull. Am. Math. Soc.,76, 887–993 (1970).
P. Halmos, “Some unsolved problems of unknown depth about operators on Hilbert space,” Proc. R. Soc. Edinburgh,76A, 67–76 (1976).
B. Johnson and J. Williams, “The range of normal derivations,” Pac. J. Math.,58, 105–122 (1975).
T. Kato, “Smooth operators and commutators,” Stud. Math.,31, 535–546 (1968).
C. Putnam, Commutation Properties of Hilbert Space Operators and Related Topics, Springer-Verlag (1967).
B. Simon, Trace Ideals and Their Applications, London Math. Soc., Lecture Notes,35, Cambridge (1979).
J. Stampfly, “Hyponormal operators,” Pac. J. Math.,12, 1453–1458 (1962).
D. Voiculescu, “Some extensions of quasitriangularity,” Rev. Roum. Math. Pures Appl.,18, 1303–1320 (1973).
D. Voiculescu, “Some results on norm-ideal perturbations of Hilbert space operators,” J. Operator Theory,2, 3–37 (1973).
G. Weiss, “The Fuglede commutativity theorem modulo Hilbert-Schmidt class and generating functions for matrix operators,” Trans. Am. Math. Soc.,246, 193–210 (1978).
G. Weiss, “The Fuglede commutativity modulo Hilbert-Schmidt class and generating functions for matrix operators. II,” J. Operator Theory,5, 3–16 (1981).
J. Williams, “Derivation ranges: open problems,” in: Topics in modern Operator Theory. Fifth International Conference on Operator Theory, Timiscara and Herculane, June 2–12, 1980, Basel, etc (1981), pp. 319–328.
Ho Yang, “Commutants and derivation ranges,” Tohoku Math. J.,27, 509–514 (1975).
Additional information
Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskgo Instituta im. V. A. Steklova AN SSSR, Vol. 135, pp. 182–194, 1984.
The author is profoundly grateful to V. I. Lomonosov, V. V. Peller, M. Z. Solomyak, and Yu. T. Turovskii for helpful discussions of the questions considered in the paper and for information on the results of [1, 8, 11, 14].
Rights and permissions
About this article
Cite this article
Shul'man, V.S. Multiplication operators and traces of commutators. J Math Sci 31, 2749–2757 (1985). https://doi.org/10.1007/BF02107262
Issue Date:
DOI: https://doi.org/10.1007/BF02107262