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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
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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.

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

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