The uuniqueness of the symmetric structure in ideals of compact operators

Автор: Aminov Behzod Rasulovich, Chilin Vladimir Ivanovich

Журнал: Владикавказский математический журнал @vmj-ru

Статья в выпуске: 1 т.20, 2018 года.

Бесплатный доступ

Let H be a separable infinite-dimensional complex Hilbert space, let L(H) be the C∗-algebra of bounded linear operators acting in H, and let K(H) be the two-sided ideal of compact linear operators in L(H). Let (E,∥⋅∥E) be a symmetric sequence space, and let CE:={x∈K(H):{sn(x)}∞n=1∈E} be the proper two-sided ideal in L(H), where {sn(x)}∞n=1 are the singular values of a compact operator x. It is known that CE is a Banach symmetric ideal with respect to the norm ∥x∥CE=∥{sn(x)}∞n=1∥E. A symmetric ideal CE is said to have a unique symmetric structure if CE=CF, that is E=F, modulo norm equivalence, whenever (CE,∥⋅∥CE) is isomorphic to another symmetric ideal (CF,∥⋅∥CF). At the Kent international conference on Banach space theory and its applications (Kent, Ohio, August 1979), A. Pelczynsky posted the following problem: (P) Does every symmetric ideal have a unique symmetric structure? This problem has positive solution for Schatten ideals Cp, 1≤p

Еще

Symmetric ideal of compact operators, uniqueness of a symmetric structure, positive isometry

Короткий адрес: https://sciup.org/143162465

IDR: 143162465   |   DOI: 10.23671/VNC.2018.1.11394

Список литературы The uuniqueness of the symmetric structure in ideals of compact operators

  • Abramovich Yu. A. Isometries of norm latties, Convex Anal. and Related Problems, 1988, vol. 43 (60), pp. 74-80.
  • Aminov B. R., Chilin V. I. Isometries of perfect norm ideals of compact operators, Studia Math., 2018, vol. 241 (1), pp. 87-99 DOI: 10.4064/sm170306-19-4
  • Arazy J., Lindenstrauss J. Some linear topological properties of the spaces Cp of operators on Hilbert space, Composite Math., 1975, vol. 30, pp. 81-111 DOI: 10.1016/0022-1236(81)90052-5
  • Arazy J. Basic sequences, embeddings, and the uniqueness of the symmetric structure in unitary matrix spaces, J. Func. Anal., 1981, vol. 40, pp. 302-340.
  • Bennet C., Sharpley R. Interpolation of Operators. Acad. Press, INC, 1988, 483 p.
  • Bratteli O., Robinson D. W. Operator Algebras and Quantum Statistical Mechaniks. N.Y.-Heidelber-Berlin: Springer-Verlag, 1979.
  • Carothers N., Dilworth S. Subspaces of Lp,q, Proc. Amer. Math. Soc., 1988, vol. 104, no. 2, pp. 537-545.
  • Chilin V. I., Medzhitov A. M., and Sukochev F. A. Isometries of non-commutative Lorentz spaces, Math. Z., 1989, vol. 200, pp. 527-545.
  • Chilin V., Krygin A., and Sukochev F. Extreme points of convex fully symmetric sets of measurable operators, Integr. Equat. Oper. Theory, 1992, vol. 15, pp. 186-226.
  • Chilin V. I., Sadovskaya O. V. Isomorphic classification of spaces of Lorentz sequences, Uzbek Math. J., 2017, no. 3m pp. 169-173.
  • Dodds P. G., Dodds T. K., and Pagter B. Noncommutative Kothe duality, Trans. Amer. Math. Soc., 1993, vol. 339 (2), pp. 717-750 DOI: 10.2307/2154295
  • Gohberg I. C., Krein M. G. Introduction to the Theory of Linear Nonselfadjoint Operators. Providence, RI: Amer. Math. Soc., 1969 (Translations of Math. Monographs; vol. 18).
  • Lewis D. R. An isomorphic characterization of the Schmidt class, Composito Mathematica, 1975, vol. 30 (3), pp. 293-297.
  • Lord S., Sukochev F., and Zanin D. Singular Traces. Theory and Applications. Berlin/Boston: Walter de Gruyter GmbH, 2013.
  • Simon B. Trace Ideals and their Applications/2nd ed. Providence, RI: Amer. Math. Soc., 2005 (Math. Surveys and Monographs; vol. 120).
  • Sourour A. Isometries of norm ideals of compact operators, J. Funct. Anal., 1981. vol. 43, pp. 69-77 DOI: 10.1016/0022-1236(81)90038-0
Еще
Статья научная