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Research articles

Traces of Hilbert space operators and their recent history

Pages 623-649 | Received 17 Dec 2018, Published online: 17 Apr 2019

References

  • A. Albeverio, D. Guido, A. Ponosov, and S. Scarlatti, Singular traces and compact operators, J. Funct. Anal. 137 (1996), 281–302. doi: 10.1006/jfan.1996.0047
  • A. Böttcher and A. Pietsch, Orthogonal and skew-symmetric operators in real Hilbert space, Integr. Equ. Oper. Theory 74 (2012), 497–511. doi: 10.1007/s00020-012-1999-z
  • J.W. Calkin, Two-sided ideals and congruences in the ring of bounded operators in Hilbert space, Ann. of Math. 42 (1941), 839–873. doi: 10.2307/1968771
  • A. Connes, Noncommutative Geometry, Acad. Press, New York etc., 1994.
  • A. Connes and H. Moscovici, The local index formula in noncommutative geometry, Geom. Funct, Anal. 5 (1995), 174–243. doi: 10.1007/BF01895667
  • J. Dixmier, Les algèbres d’opérateurs dans l’espace hilbertien (Algèbres de von Neumann), Gauthier–Villars, Paris, 1957.
  • J. Dixmier, Existence de traces non normales, C. R. Acad. Sci. Paris Sér. A-B 262 (1966), 1107–1108.
  • P.G. Dodds, B. de Pagter, E.M. Semenov, and F.A Suchochev, Symmetric functionals and singular traces, Positivity 2 (1998), 47–75. doi: 10.1023/A:1009720826217
  • K. Dykema, T. Figiel, G. Weiss, and M. Wodzicki, Commutator structure of operator ideals, Adv. in Math. 185 (2004), 1–79. doi: 10.1016/S0001-8708(03)00141-5
  • K. Dykema and N. Kalton, Spectral characterization of of sums of commutators II, J. Reine Angew. Math. 504 (1998), 127–137.
  • J.A. Erdős, On the trace of a trace class operator, Bull. London Math. Soc. 6 (1974), 47–50. doi: 10.1112/blms/6.1.47
  • T. Figiel, Lectures at the GDR-Polish Functional Analysis Seminar, Georgenthal, April 1986, unpublished.
  • T. Figiel and N. Kalton, Symmetric linear functionals on function spaces, In: Function Spaces, Interpolation Theory and Related Topics, M. Cwikel, M. Engliš, A. Kufner, L. Persson and G. Sparr, (eds.), De Gruyter & Co., Berlin, 2002.
  • I.C. Gohberg and M.G. Kreĭn, Introduction to the Theory of Nonselfadjoint Operators in Hilbert Space, Amer. Math. Soc., Providence, 1969; Russian original: Nauk, Moscow, 1965.
  • J.M. Gracia-Bondía, J.C. Várilly, and H. Figueroa, Elements of Noncommutative Geometry, Birkhäuser, Boston/Basel/Berlin, 2001.
  • B. Gramsch, Integration und holomorphe Funktionen in lokalbeschränkten Räumen, Math. Ann. 162 (1965), 190–210. doi: 10.1007/BF01361943
  • B. Gramsch, Transformationen in lokalbeschränkten Vektorräumen, Math. Ann. 165 (1966), 135–151. doi: 10.1007/BF01344009
  • A. Grothendieck, Produits tensoriels topologiques et espaces nucléaires, Memoirs Amer. Math. Soc. Vol. 16, Providence, RI, 1955 (Thesis, Nancy 1953).
  • A. Guichardet, La trace de Dixmier et autres traces, L’Enseignement Math. (2) 61 (2015), 461–481. doi: 10.4171/LEM/61-3/4-8
  • A. Horn, On the eigenvalues of a matrix with prescribed singular values, Proc. Amer. Math. Soc. 5 (1954), 4–7. doi: 10.1090/S0002-9939-1954-0061573-6
  • N. Kalton, Unusual traces on operator ideals, Math. Nachr. 134 (1987), 119–130. doi: 10.1002/mana.19871340108
  • N. Kalton, Spectral characterization of of sums of commutators I, J. Reine Angew. Math. 504 (1998), 115–125.
  • N. Kalton and F.A. Sukochev, Symmetric norms and spaces of operators, J. Reine Angew. Math. 621 (2008), 81–121.
  • T. Kato, Perturbation theory for nullity, deficiency and other quantities of linear operators, J. Analyse Math. 6 (1958), 261–322. doi: 10.1007/BF02790238
  • G. Levitina, A. Pietsch, F.A. Sukochev, and D. Zanin, Completeness of quasinormed operator ideals generated by s-numbers, Indag. Math. (New Series) 25 (2014), 49–58. doi: 10.1016/j.indag.2013.07.005
  • V.B. Lidskiĭ, Nonselfadjoint operators with a trace, Amer. Math. Soc. Transl. (2), 47 (1965), 43–46; Russian original: Doklady Akad. Nauk SSSR 25 (1959), 485–487.
  • S. Lord, A. Sedaev, and F.A. Sukochev, Dixmier traces as singular symmetric functionals and applications to measurable operators, J. Funct. Anal. 224 (2005), 72–106. doi: 10.1016/j.jfa.2005.01.002
  • S. Lord, F. Sukochev, and D. Zanin, Singular Traces, De Gruyter, Berlin, 2013.
  • S. Lord, F. Sukochev, and D. Zanin, Advances in Dixmier traces and applications, preprint.
  • J. von Neumann, Mathematische Grundlagen der Quantenmechanik, Springer, Berlin, 1932.
  • F. Nicola and L. Rodino, Global Pseudo-Differential Calculus on Euclidean Spaces, Birkhäuser, Basel, 2010.
  • A. Pietsch, Operator Ideals, Deutsch. Verlag Wiss., Berlin, 1978; North–Holland, Amsterdam/London/New York/Tokyo, 1980.
  • A. Pietsch, Eigenvalues and s-Numbers, Geest&Portig, Leipzig, and Cambridge Univ. Press, Cambridge, 1987.
  • A. Pietsch, Traces and shift invariant functionals, Math. Nachr. 145 (1990), 7–43. doi: 10.1002/mana.19901450102
  • A. Pietsch, History of Banach Spaces and Linear Operators, Birkhäuser, Boston, 2007.
  • A. Pietsch, Dixmier traces of operators on Banach and Hilbert spaces, Math. Nachr. 285 (2012), 1999–2028. doi: 10.1002/mana.201100137
  • A. Pietsch, Shift-invariant functionals on Banach sequence spaces, Studia Math. 214 (2013), 37–66. doi: 10.4064/sm214-1-3
  • A. Pietsch, Connes–Dixmier versus Dixmier traces, Integr. Equ. Oper. Theory 77 (2013), 243–259. doi: 10.1007/s00020-013-2056-2
  • A. Pietsch, Traces on operator ideals and related linear forms on sequence ideals (part I ), Indag. Math. (New Series) 25 (2014), 341–365. doi: 10.1016/j.indag.2012.08.008
  • A. Pietsch, Traces on operator ideals and related linear forms on sequence ideals (part II ), Integr. Equ. Oper. Theory 79 (2014), 255–299. doi: 10.1007/s00020-013-2114-9
  • A. Pietsch, Traces of operators and their history, Acta Comment. Univ. Tartu. Math. 18 (2014), 51–64.
  • A. Pietsch, Traces on operator ideals and related linear forms on sequence ideals (part III ), J. Math. Anal. Appl. 421 (2015), 971–981. doi: 10.1016/j.jmaa.2014.07.069
  • A. Pietsch, A new approach to operator ideals on Hilbert space and their traces, Integr. Equ. Oper. Theory 89 (2017), 595–606. doi: 10.1007/s00020-017-2410-x
  • A. Pietsch, The spectrum of shift operators and the existence of traces, Integr. Equ. Oper. Theory (2018), 90:17, https://doi.org/10.1007/s00020-018-2427-9.
  • A. Pietsch, Traces and symmetric linear forms, Arch. Math. (Basel), 112 (2019), 83–92. doi: 10.1007/s00013-018-1261-2
  • A. Pietsch, A new view at Dixmier traces on ℒ1,∞(H), Integr. Equ. Oper. Theory (2019), 91:21, https://doi.org/10.1007/s00020-019-2509-3.
  • W.C. Ridge, Approximate point spectrum of a weighted shift, Trans. Amer. Math. Soc. 147 (1970), 349–356. doi: 10.1090/S0002-9947-1970-0254635-5
  • D. Robert, On the traces of operators (from Grothendieck to Lidskii), EMS News Letter September (2017), 26–33.
  • R. Schatten, Norm Ideals of Completely Continuous Operators, Springer, Heidelberg/Berlin/New York, 1960.
  • A.A. Sedaev, E.M. Semenov, and F.A. Sukochev, Fully symmetric function spaces without an equivalent Fatou norm, Positivity 19 (2015), 419–437. doi: 10.1007/s11117-014-0305-5
  • A.A. Sedaev, F.A. Sukochev, Dixmier measurability in Marcinkiewicz spaces and applications, J. Funct. Anal. 265 (2013), 3053–3066. doi: 10.1016/j.jfa.2013.08.014
  • E. Semenov, F. Sukochev, and A. Usachev, Banach limits and traces on ℒ1,∞, Adv. Math. 285 (2015), 568–628. doi: 10.1016/j.aim.2015.08.010
  • B. Simon, Trace Ideals and their Applications, London Math. Soc. Lecture Note, Vol. 35, Cambridge Univ. Press, Cambridge, 1979.
  • F. Sukochev, Completeness of quasi-normed symmetric operator spaces, Indag. Math. (New Series) 25 (2014), 376–388. doi: 10.1016/j.indag.2012.05.007
  • F. Sukochev, A. Usachev, and D. Zanin, On the distinction between the classes of Dixmier and Connes–Dixmier traces, Proc. Am. Math. Math. 141 (2013), 2169–2179. doi: 10.1090/S0002-9939-2012-11853-2
  • F. Sukochev and D. Zanin, Which traces are spectral? Adv. in Math. 252 (2014), 406–428. doi: 10.1016/j.aim.2013.10.028
  • J.V. Varga, Traces on irregular ideals, Proc. Amer. Math. Soc. 107 (1989), 715–723. doi: 10.1090/S0002-9939-1989-0984818-8
  • J.V. Varga, Traces and commutators of ideals of compact operators, unpublished Dissertation, The Ohio State Univ., 1995.
  • M. Väth, The dual space of L∞ is L1, Indag. Math. (New Series) 9 (1998), 619–625. doi: 10.1016/S0019-3577(98)80039-6
  • H. Weyl, Inequalities between the two kinds of eigenvalues of a linear transformation, Proc. Nat. Acad. Sci. USA 35 (1949), 408–411. doi: 10.1073/pnas.35.7.408
  • M. Wodzicki, Vestigia investiganda, Mosc. Math. J. 2 (2002), 769–798. doi: 10.17323/1609-4514-2002-2-4-769-798

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