1997/09/08 by N. J. Kalton, Nigel J. Kalton, Kalton, Nigel J.
Mathematics · #47B10 #Advanced Operator Algebra Research #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Spectral Theory in Mathematical Physics #math.FA #msc:47B10
paper · pdf · doi:10.48550/arxiv.math/9709209
arxiv created 1997/09/08 · openalex publication_date 1997/09/08 · arxiv updated 2016/09/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Suppose \Cal J is a two-sided quasi-Banach ideal of compact operators on a separable infinite-dimensional Hilbert space \Cal H. We show that an operator T∈\Cal J can be expressed as finite linear combination of commutators [A,B] where A∈\Cal J and B∈\Cal B(\Cal H) if and only its eigenvalues (λn) (arranged in decreasing order of absolute value, repeated according to algebraic multiplicity and augmented by zeros if necessary) satisfy the condition that the diagonal operator \diag\\frac1n(λ1+⋯ +λn)\ is a member of \Cal J. This answers (for quasi-Banach ideals) a question raised by Dykema, Figiel, Weiss and Wodzicki.