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 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.