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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On zero-diagonal operators and traces
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by Peng Fan, Che Kao Fong and Domingo A. Herrero
Proc. Amer. Math. Soc. 99 (1987), 445-451
DOI: https://doi.org/10.1090/S0002-9939-1987-0875378-9

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 $|T - {A_k}{|_\mathcal {J}} \to 0$ ($\mathcal {J}$ 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
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Bibliographic Information
  • © Copyright 1987 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 99 (1987), 445-451
  • MSC: Primary 47A12; Secondary 47B10
  • DOI: https://doi.org/10.1090/S0002-9939-1987-0875378-9
  • MathSciNet review: 875378