Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On zero-diagonal operators and traces

Authors: Peng Fan, Che Kao Fong and Domingo A. Herrero
Journal: Proc. Amer. Math. Soc. 99 (1987), 445-451
MSC: Primary 47A12; Secondary 47B10
MathSciNet review: 875378
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A Hilbert space operator $ A$ is called zero-diagonal if there exists an orthonormal basis $ \phi = \{ {e_j}\} _{j = 1}^\infty $ such that $ \left\langle {A{e_j},{e_j}} \right\rangle = 0$ for all $ j$. It is known that $ T$ is the norm limit of a sequence $ \{ {A_k}\} $ of zero-diagonal operators iff $ 0 \in {W_e}(T)$, the essential numerical range of $ T$. Our first result says that if $ 0 \in {W_e}(T)$ and $ \mathcal{J}$ is an ideal of compact operators strictly larger than the trace class, then the sequence $ \{ {A_k}\} $ can be chosen so that $ \vert T - {A_k}{\vert _\mathcal{J}} \to 0$ ( $ \mathcal{J}$ cannot be replaced by the trace class!). If $ A$ is zero-diagonal, then the series $ \sum _{j = 1}^\infty \left\langle {A{e_j},{e_j}} \right\rangle $ converges to a value (zero) that can be called "the trace of $ A$ with respect to the basis $ \phi $". Our second result provides, for each operator $ T$, the structure of the set of all possible "traces" of $ T$ (in the above sense). In particular, this set is always either the whole complex plane, a straight line, a singleton, or the empty set.

References [Enhancements On Off] (What's this?)

  • [1] J. H. Anderson, Derivations, commutators and the essential numerical range, Thesis, Indiana University, 1971.
  • [2] A. Ben-Artzi, Traces of compact operators, Integral Equations Operator Theory 7 (1984), 310-324. MR 756762 (86g:47018)
  • [3] P. Fan, On the diagonal of an operator, Trans. Amer. Math. Soc. 283 (1984), 239-251. MR 735419 (86b:47034)
  • [4] P. A. Fillmore, J. G. Stampfli, and J. P. Williams, On the essential numerical range, the essential spectrum and a problem of Halmos, Acta Sci. Math. (Szeged) 33 (1972), 179-192. MR 0322534 (48:896)
  • [5] P. R. Halmos, Finite-dimensional vector spaces, Van Nostrand, Princeton, N. J., 1958. MR 0089819 (19:725b)
  • [6] -, A Hilbert space problem book, Van Nostrand, Princeton, N. J., 1967. MR 0208368 (34:8178)
  • [7] D. A. Herrero, An essay on quasitriangularity, Proc. 11th Internat. Conf. on Operator Theory, Bucharest 1986 (Romania), Operator Theory: Advances and Applications, Birkhäuser-Verlag, Basel (to appear). MR 942918 (89h:47028)
  • [8] T. Kato, Perturbation theory for linear operators, Springer-Verlag, New York, 1966. MR 0203473 (34:3324)
  • [9] R. Schatten, Norm ideals of completely continuous operators, Springer-Verlag, Berlin, 1960. MR 0119112 (22:9878)
  • [10] Q. F. Stout, Schur products of operators and the essential numerical range, Trans. Amer. Math. Soc. 264 (1981), 39-47. MR 597865 (82h:47029)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 47A12, 47B10

Retrieve articles in all journals with MSC: 47A12, 47B10

Additional Information

Keywords: Zero-diagonal, essential numerical range, trace, set of traces, trace class operators
Article copyright: © Copyright 1987 American Mathematical Society

American Mathematical Society