Abstract
We offer a new perspective and some advances on the 1971 Pearcy-Topping problem: is every compact operator a commutator of compact operators? Our goal is to analyze and generalize the 1970’s work in this area of Joel Anderson. We reduce the general problem to a simpler sequence of finite matrix equations with norm constraints, while at the same time developing strategies for counterexamples. Our approach is to ask which compact operators are commutators AB − BA of compact operators A,B; and to analyze the implications of Joel Anderson’s contributions to this problem. By extending the techniques of Anderson, we obtain new classes of operators that are commutators of compact operators beyond those obtained by the second and the fourth author. We also found obstructions to extending Anderson’s techniques to obtain any positive compact operator as a commutator of compact operators. Some of these constraints involve general block-tridiagonal matrix forms for operators and some involve B(H)-ideal constraints. Finally, we provide some necessary conditions for the Pearcy-Topping problem involving singular numbers and B(H)-ideal constraints.
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Notes
Note that this is essentially a discrete differential inequality since
$$\begin{aligned} \frac{s_n - s_{n-1}}{n - (n-1)} = k_n > \delta s_n. \end{aligned}$$Since \(\Sigma ({\mathcal {J}})\) is hereditary, it is closed under 2-ampliation if and only if it is closed under k-ampliation for any \(k \ge 2\).
The sequence \(s(A_n)\) is finite with length \(\min \{k_n, k_{n+1}\}\); but we may pad it with zeros to make it have length \(k_n\) in case \(k_{n+1} < k_n\).
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Sasmita Patnaik: Supported by Science and Engineering Research Board, Core Research Grant 002514
Gary Weiss: Partially supported by Simons Foundation collaboration grants 245014 and 636554
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Loreaux, J., Patnaik, S., Petrovic, S. et al. On Commutators of Compact Operators: Generalizations and Limitations of Anderson’s Approach. Integr. Equ. Oper. Theory 96, 16 (2024). https://doi.org/10.1007/s00020-024-02764-9
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DOI: https://doi.org/10.1007/s00020-024-02764-9