Abstract.
This is a historical survey that includes a progress report on the 1971 seminal paper of Pearcy and Topping and 32 years of subsequent investigations by a number of researchers culminating in a completely general characterization, for arbitrary ideal pairs, of their commutator ideal in terms of arithmetic means.
This characterization has applications to the study of generalized traces (linear functionals vanishing on the commutator ideal [I,B(H)]) and to the study of the B(H)-ideal lattice and certain special sublattices. The structure of commutator ideals is essential for investigating traces which in turn is relevant for the calculation of the cyclic homology and the algebraic K-theory of operator ideals.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
J. Anderson, Commutators of Compact Operators. J. Reine Angew. Math. 291 (1977), 128–132.
J. Anderson, L.N. Vaserstein, Commutators in Ideals of Trace Class Operators. Indiana University Mathematics Journal 35(2) (1986), 345–372.
J. Anderson, Commutators in Ideals of Trace Class Operators. II, Indiana University Mathematics Journal 35(2) (1986), 373–378.
L.G. Brown, Traces of Commutators of Schatten-von Neumann Class Operators. J. Reine Angew. Math. 451 (1994), 171–174.
A. Brown, C. Pearcy, Structure of Commutators of Operators, Ann. of Math. (2) 82 (1965), 112–127.
J.M. Calkin, Two-Sided Ideals and Congruences in the Ring of Bounded Operators in Hilbert Space, Ann. of Math. (2) 42 (1941), 839–873.
J. Dixmier, Existence de Traces non Normales. C. R. Acad. Sci. Paris Sér. A-B 262 (1966), A1107–A1108.
K. Dykema, T. Figiel, G. Weiss, M. Wodzicki, The Commutator Structure of Operator Ideals, Adv. Math. 185 (2004), no. 1, 1–79.
K.J. Dykema, N.J. Kalton, Spectral Characterization of Sums of Commutators. II, J. Reine Angew. Math. 504 (1998), 127–137.
K. Dykema, G. Weiss, M. Wodzicki, Unitarily Invariant Trace Extensions Beyond the Trace Class, Complex analysis and related topics (Cuernavaca, 1996), Oper. Theory Adv. Appl. 114 (2000), 59–65.
K. Fan, Maximum Properties and Inequalities for the Eigenvalues of Completely Continuous Operators. Proc. Natl. Acad. Sci. USA 37 (1951), 760–766.
C.K. Fong, C.R. Miers, A.R. Sourour, Lie and Jordan Ideals of Operators on Hilbert Space. Proc. Amer. Math. Soc. 84 (1982), no. 4, 516–520.
I.C. Gohberg, M.G. Kreĭn, Introduction to the Theory of Linear Nonselfadjoint Operators. American Mathematical Society (1969).
P.R. Halmos, Commutators of Operators. II, Amer. J. Math. 76 (1954), 191–198.
P.R. Halmos, A Hilbert Space Problem Book, 2nd Edition, Graduate Texts in Mathematics (19), Springer-Verlag (1982). From 1st Edition, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London (1967).
G.H. Hardy, J.E. Littlewood, G. Pólya, Inequalities. Cambridge University Press (1934, 1952, 1988).
W. Heisenberg, Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik, Zeitschrift für Physik 43 (1927), 172–198.
V. Kaftal, G. Weiss, Traces, Ideals, and Arithmetic Means. Proc. Natl. Acad. Sci. USA 99(11) (2002), 7356–7360.
N.J. Kalton, Trace-Class Operators and Commutators. J. Funct. Anal. 86 (1989), 41–74.
N.J. Kalton, Spectral Characterization of Sums of Commutators. I, J. Reine Angew. Math. 504 (1998), 115–125.
C. Laurie, E. Nordgren, H. Radjavi, P. Rosenthal, On Triangularization of Algebras of Operators, Journal für die Reine und Angewandte Mathematik 327 (1981), 143–155.
V.I. Macaev, A class of Completely Continuous Operators. Dokl. Akad. Nauk SSSR 139 (1961), 548–551.
A.W. Marshall, I. Olkin, Inequalities: Theory of Majorization and its Applications, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], Mathematics in Science and Engineering 143 (1979).
C. Pearcy, D. Topping, Commutators and Certain II 1 -factors. J. Functional Analysis 3 (1969), 69–78.
C. Pearcy, D. Topping, On Commutators in Ideals of Compact Operators. Michigan Math. J. 18 (1971), 247–252.
J. Varga, Traces on Irregular Ideals. Proc. Amer. Math. Soc. 1073 (1989), 715–723.
G. Weiss, Commutators and Operators Ideals. dissertation (1975), University of Michigan Microfilm.
G. Weiss, Commutators of Hilbert-Schmidt Operators. II. Integral Equations Operator Theory 3(4) (1980), 574–600.
G. Weiss, Commutators of Hilbert-Schmidt Operators. I. Integral Equations Operator Theory 9 (1986), 877–892.
G. Weiss, Classification of Certain Commutator Ideals and the Tensor Product Closure Property. Integral Equations Operator Theory 12 (1989), 99–128.
H. Wielandt, Über die Unbeschränktheit der Operatoren der Quantenmechanik. Math. Ann. 121 (1949) 21.
A. Wintner, The Unboundedness of Quantum-Mechanical Matrices. Phys. Rev. 71 (1947), 738–739.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2004 Birkhäuser Verlag Basel/Switzerland
About this paper
Cite this paper
Weiss, G. (2004). B(H)-Commutators: A Historical Survey. In: Gaşpar, D., Timotin, D., Zsidó, L., Gohberg, I., Vasilescu, FH. (eds) Recent Advances in Operator Theory, Operator Algebras, and their Applications. Operator Theory: Advances and Applications, vol 153. Birkhäuser Basel. https://doi.org/10.1007/3-7643-7314-8_20
Download citation
DOI: https://doi.org/10.1007/3-7643-7314-8_20
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-7643-7127-2
Online ISBN: 978-3-7643-7314-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)