Quasicentral Modulus and Self-similar Sets: A Supplementary Result to Voiculescu’s Work

In his recent work, Voiculescu generalized his remarkable formula for the quasicentral modulus of a commuting n -tuple of hermitian operators with respect to the ( n , 1)-Lorentz ideal to the case where its spectrum is contained in a Cantor-like self-similar set in a certain class. In this note, we...

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Bibliographic Details
Published inIntegral equations and operator theory Vol. 95; no. 2
Main Authors Ikeda, Kozo, Izumi, Masaki
Format Journal Article
LanguageEnglish
Published Cham Springer International Publishing 01.06.2023
Springer Nature B.V
Subjects
Analysis
Mathematics
Mathematics and Statistics
Original Research
Self-similarity
Upper bounds
Hausdorff measure
Quasicentral modulus
Lorentz
Secondary 28A78
Primary 47A55
28A80
47B10
normed ideal
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Summary:In his recent work, Voiculescu generalized his remarkable formula for the quasicentral modulus of a commuting n -tuple of hermitian operators with respect to the ( n , 1)-Lorentz ideal to the case where its spectrum is contained in a Cantor-like self-similar set in a certain class. In this note, we treat general self-similar sets satisfying the open set condition, and obtain lower and upper bounds of the quasicentral modulus. Our proof shows that Voiculescu’s formula holds for a class of self-similar sets including the Sierpinski gasket and the Sierpinski carpet.
ISSN:0378-620X
1420-8989
DOI:10.1007/s00020-023-02734-7