Perturbations of operators and non-commutative condensers, an update on the quasicentral modulus

TL;DR

The work updates multivariable perturbation theory via the quasicentral modulus $k_{\mathscr J}(\tau)$ for $n$-tuples of operators within normed ideals, deriving sharp analogues of one-variable perturbation results for $n\ge 3$ and exploring invariance under perturbations modulo $\boldsymbol{\mathscr C^-_n}$. It provides exact formulas for commuting $n$-tuples, a hybrid modulus, and a fractal-spectrum case, linking these operator-theoretic quantities to a noncommutative analogue of condenser capacity from nonlinear potential theory. A central contribution is the noncommutative dictionary that connects condenser capacity, gradients, and rearrangement-invariant norms to their operator-theoretic counterparts, enabling variational problems and graph-based extensions, including discrete groups and Cayley graphs. The paper also investigates entropy connections via $k^-_\infty$ and $\mathscr H_P(\theta)$, discusses open problems in the $n=2$ case, and outlines semifinite generalizations, thereby broadening the reach of noncommutative potential-theoretic methods in perturbation theory and noncommutative geometry.

Abstract

This is an update on the quasicentral modulus, an invariant for an n-tuple of Hilbert space operators and a rearrangement invariant norm, that plays a key-role in sharp multivariable generalizations of the classical Weyl-von Neumann-Kuroda and Kato-Rosenblum theorems of perturbation theory. There are also connections with self-similar measures on certain fractals and to the Kolmogorov-Sinai dynamical entropy. Some open problems are also pointed out. Recently a non-commutative analogy with condenser capacity in nonlinear potential theory is emerging, that provides a new perspective on the subject.