Commutator structure of operator ideals

Advances in Mathematics

Volumn 185, Issue 1, 2004, Pages 1-79

  Dykema, Ken ^a   Figiel, Tadeusz ^b   Weiss, Gary ^c   Wodzicki, Mariusz ^d  

^a TEXAS A AND M UNIVERSITY   (United States)

^b INSTITUTE OF MATHEMATICS   (Poland)

^c UNIVERSITY OF CINCINNATI   (United States)

^d UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 2942537946     PISSN: 00018708     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0001-8708(03)00141-5     Document Type: Article

References (73)

1. S. Aljančić, D. Arandelović, O-regularly varying functions, Publ. Inst. Math. Beograd (N.S.) 22(36) (1977) 5-22. MR 57#6317.
2. G.D. Allen, L.C. Shen, On the structure of principal ideals of operators, Trans. Amer. Math. Soc. 238 (1978) 253-270. MR 57#7230.
3. J.H. Anderson, Commutators of compact operators, J. Reine Angew. Math. 291 (1977) 128-132. MR 56#1122.
4. J.H. Anderson, Commutators in ideals of trace class operators II, Indiana Univ. Math. J. 35 (1986) 373-378. MR 87f.47025b.
5. J.H. Anderson, J.G. Stampfli, Commutators and compressions, Israel J. Math. 10 (1971) 433-441. MR 47#874.
6. W. Banaszczyk, The Steinitz constant of the plane, J. Reine Angew. Math. 373 (1987) 218-220. MR 88e:52016.
7. W. Banaszczyk, The Steinitz theorem on rearrangement of series for nuclear spaces, J. Reine Angew. Math. 403 (1990) 187-200. MR 90k:46010.
8. W. Banaszczyk, A note on the Steinitz constant of the Euclidean plane, C. R. Math. Rep. Acad. Sci. Canada 12 (1990) 97-102. MR 91g:52005.
9. C. Bennett, R. Sharpley, Interpolation of Operators, in: Pure and Applied Mathematics, Vol. 129, Academic Press, New York, 1988.
10. J. Bergh, J. Löfström, Interpolation Spaces : An Introduction, Grundlehren Math. Wiss., Vol. 223, Berlin, New York: Springer-Verlag, 1976.
11. N.H. Bingham, C.M. Goldie, J.L. Teugels, Regular Variation, Cambridge University Press, Cambridge, 1987. MR 88i:26004.
12. R. Bojanić, E. Seneta, A unified theory of regularly varying sequences, Math. Z. 134 (1973) 91-106. MR 48#11407 26A12.
13. N. Bourbaki, Éléments de mathématique, Fasc. XXXII: Théories spectrales, Hermann, Paris, 1967. MR 35#4725.
14. D.W. Boyd, Indices of function spaces and their relationship to interpolation, Canad. J. Math. 21 (1969) 1245-1254. MR 54#909.
15. D.W. Boyd, Indices for the Orlicz spaces, Pacific J. Math. 33 (1971) 315-323. MR 46#6008.
16. A. Brown, C. Pearcy, Structure of commutators of operators, Ann. of Math. 82(2) (1965) 112-127. MR 31#2612.
17. A. Brown, C. Pearcy, N. Salinas, Ideals of compact operators on Hilbert space, Michigan Math. J. 18 (1971) 373-384. M R 45#909.
18. L.G. Brown, Traces of commutators of Schatten-von Neumann class operators, J. Reine Angew. Math. 451 (1994) 171-174. MR 95b:47019.
19. J.W. Calkin, Two-sided ideals and congruences in the ring of bounded operators in Hilbert space, Ann. of Math. 42(2) (1941) 839-873. MR 3,208c.
20. A. Connes, Noncommutative differential geometry, Part I, the Chern character in K-homology, Inst. Hautes Etudes Sci., Bures-sur-Yvette, 1982.
21. A. Connes, Noncommutative differential geometry, Part II, de Rham homology and noncommutative algebra, Inst. Hautes Etudes Sci., Bures-sur-Yvette, 1983.
22. A. Connes, Noncommutative differential geometry, Publ. IHES 62 (1985) 41-144. MR 87i: 58162.
23. A. Connes, Caractères des représentations θ-sommables des groupes discrets, C. R. Acad. Sci. Paris Sér. I Math. 312 (1991) 661-666. MR 92c:19003.
24. A. Connes, Noncommutative Geometry, Academic Press, New York, 1994.
25. P.W. Day, Rearrangement inequalities, Canad. J. Math. 24 (1972) 930-943. MR 46#9258.
26. J. Dixmier, Existence de traces non normale, C. R. Acad. Sci. Paris Sér. A-B 262 (1966) A1107-1108. MR 33#4695.
27. K. Dykema, N.J. Kalton, Spectral characterization of sums of commutators. II, J. Reine Angew. Math. 504 (1998) 127-137. MR 99g:47079.
28. K. Dykema, T. Figiel, G. Weiss, M. Wodzicki, Commutator structure of operator ideals, IMADA preprint PP-1997-22, May 13, 1997, Odense, 28pp.
29. K. Dykema, G. Weiss, M. Wodzicki, Unitarily invariant trace extensions beyond the trace class, in: Complex Analysis and Related Topics (Cuernavaca, 1996), Oper. Theory. Adv. Appl. 114 (2000) 59-65. MR 2001c:47025.
30. Ky Fan, Maximum properties and inequalities for the eigenvalues of completely continuous operators, Proc. Natl Acad. Sci. USA 37 (1951) 760-766.
31. E. Fischer, Über quadratische Formen mit reellen Koeffizienten, Monatsh. Math. Phys 16 (1905) 234-249.
32. D.J.H. Garling, On symmetric sequence spaces, Proc. London. Math. Soc. 16 (3) (1966) 85-106.
33. Foreign source 1965. (Engl. transl.: I.C. Gokhberg, M.G. Kreǐn, Introduction to the Theory of Linear Nonselfadjoint Operators, American Mathematical Society, Providence, RI, 1969). MR 36#3137 (tr.39#7447)
34. Foreign source 14:2 (1980), 56-57. (Engl. transl.: V.S. Grinberg, S.V. Sewast'janow, Regarding the value of Steinitz's constant, Funct. Anal. Appl. 14(2) (1980) 125-126) MR 81h:52008.
35. P. Halmos, Commutators of operators. II, Amer. J. Math. 76 (1954) 191-198. MR 15:568d.
36. G.H. Hardy, J.E. Littlewood, G. Pólya, Some simple inequalities satisfied by convex functions, Messenger Math. 58 (1928/29) 145-152.
37. N.J. Kalton, Unusual traces on operator ideals, Math. Nachr. 134 (1987) 119-130. MR 87a:47070.
38. N.J. Kalton, Trace-class operators and commutators, J. Funct. Anal. 86 (1989) 41-74. MR 91d:47022.
39. N.J. Kalton, Spectral characterization of sums of commutators. I, J. Reine Angew. Math. 504 (1998) 115-125. MR 99g:47078.
40. G. Köthe, O. Toeplitz, Lineare Raüme mit unendlichvielen Koordinaten und Ringe unendlicher Matrizen, J. Reine Angew. Math. 171 (1934) 193-226.
41. G.G. Lorentz, An inequality for rearrangements, Amer. Math. Monthly 60 (1953) 176-179. MR 14:626a.
42. W.A.J. Luxemburg, Rearrangement invariant Banach function spaces, Proc. Symp. Anal. Queen's Univ. 10 (1967) 83-144.
43. Foreign source (V.I. Macaev, On a class of completely continuous operators), (Dokl. Akad. Nauk SSSR) 139 (1961) 548-551. MR 24#A1617
44. W. Matuszewska, On generalized Orlicz spaces, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 8 (1960) 349-353. MR 23#A3453.
45. W. Matuszewska, Przestrzenie funkcji -całkowalnych. I. Własności ogólne -funkcji i klas funkcji -cał kowalnych (Spaces of -integrable functions. I. General properties of -functions and classes of -integrable functions), Prace Mat. 6 (1961) 121-139. MR 32#4530a.
46. W. Matuszewska, Przestrzenie funkcji -całkowainych. II. Uogólnione przestrzenie Orlicza (Spaces of -integrable functions. II. Generalized Orlicz spaces), Prace Mat. 6 (1961) 149-164. MR 32#4530b.
47. W. Matuszewska, W. Orlicz, On certain properties of -functions, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 8 (1960) 439-443. MR 23#A3454.
48. D. Morris, N. Salinas, Semiprime ideals and irreducible ideals of the ring of bounded operators on Hilbert space, Indiana Univ. Math. J. 23 (1973/74) 575-589. MR 48#4758.
49. C. Pearcy, D. Topping, On commutators in ideals of compact operators, Michigan J. Math. 18 (1971) 247-252. MR 44#2077.
50. A. Pietsch, Eigenvalues and s-numbers, Akademische Verlagsgesellschaft Geest & Portig K.-G., Leipzig, 1987; also Cambridge University Press, Cambridge, 1987. MR 88j:47022b.
51. G. Pólya, Remark on Weyl's note "Inequalities between the two kinds of eigenvalues of a linear transformation, Proc. Natl Acad. Sci. USA 36 (1950) 49-51. MR 11:526b.
52. M. Rosenblum, On the operator equation BX - XA = Q, Duke Math. J. 23 (1956) 263-269. MR 18:54d.
53. N. Salinas, Ideal sets and ideals of compact operators on Hilbert space, Indiana Univ. Math. J. 22 (1972/73) 505-521. MR 47#5642.
54. R. Schatten, Norm ideals of completely continuous operators, Ergebnisse der Mathematik und ihrer Grenzgebiete, Neue Folge, Heft, Vol. 27, Springer, Berlin, 1960. MR 22#9878.
55. R. Schatten, J. von Neumann, The cross-space of linear transformations. III, Ann. of Math. 49(2) (1948) 557-582. MR 10:256b.
56. F. Schrohe, M. Walze, J.-M. Warzecha, Construction de triplets spectraux à partir de modules de Fredholm, C. R. Acad. Sci. Paris Sér. I Math. 326 (1998) 1195-1199. MR 2000b:46104.
57. A. Schweinsberg, Principal ideals of compact operators, Indiana Univ. Math. J. 25 (1976) 229-233. MR 53#3724.
58. T. Shimogaki, Hardy-Littlewood majorants in function spaces, J. Math. Soc. Japan 17 (1965) 365-373.
59. K. Shoda, Einige Sätze über Matrizen, Japanese J. Math. 13 (1937) 361-365.
60. E. Steinitz, Bedingt konvergente Reihen und konvexe Systeme, J. Reine Angew. Math. 143 (1913) 128-175.
61. M. Tomić, Théorème de Gauss relatif au centre de gravité et son application, Bull. Soc. Math. Phys. Serbie 1 (1949) 31-40. MR 11:86a.
62. J.V. Várga, Traces on irregular ideals, Proc. Amer. Math. Soc. 107 (1989) 715-723. MR 91e:47046.
63. D. Voiculescu, Some results on norm-ideal perturbations of Hilbert space operators, J. Oper. Theory 2 (1979) 3-37. Part II, J. Oper. Theory 5 (1981) 77-100. MR 80m:47012 and MR 83f:47014.
64. D. Voiculescu, Hilbert space operators modulo normed ideals, in: Proceedings of the International Congress of Mathematicians, Warsaw, Vol. 2, 1983, PWN, Warsaw, 1984, pp. 1041-1047. MR 87b:47049.
65. D. Voiculescu, On the existence of quasicentral approximate units relative to normed ideals. I, J. Funct. Anal. 91 (1990) 1-36. MR91m:46089.
66. G. Weiss, Commutators and operator ideals, Ph.D. Thesis, University of Michigan, Ann Arbor, 1975.
67. G. Weiss, Commutators of Hilbert-Schmidt operators I, Integral Equations Oper. Theory 9 (1986) 877-892. MR 88d:47029.
68. G. Weiss, Commutators of Hilbert-Schmidt operators II, Integral Equations Oper. Theory 3/4 (1980) 574-600. MR 83h:47024.
69. H. Weyl, Inequalities between the two kinds of eigenvalues of a linear transformation, Proc. Acad. Sci. USA 35 (1949) 408-411. MR 11,37d.
70. M. Wodzicki, Algebraic K-theory and functional analysis, in: Proceedings of the First European Congress of Mathematics, Paris, Vol. II, July 6-10, 1992, pp. 485-496, Progress in Mathematics vol. 120, Birkhäuser Verlag, Basel, 1994. MR 97f.46112.
71. M. Wodzicki, Vestigia inuestiganda, Moscow Math. J. 2 (2002) 769-798.
72. M. Wodzicki, Algebraic K-theory of operator ideals, in preparation.
73. K. Zeller, Theorie der Limitierungsverfahren, Ergebnisse der Mathematik und ihrer Grenzgebiet, Neue Folge, Hefl, Vol. 15, Springer, Berlin, Göttingen, Heidelberg, 1958. MR 20#4119.