Hybrid Normed Ideal Perturbations of n-tuples of Operators I

TL;DR

This paper extends the theory of normed ideal perturbations to hybrid settings involving n-tuples of operators, introducing new techniques and generalizations for commuting hermitian operators.

Contribution

It develops a framework for hybrid normed ideal perturbations, generalizing previous methods to variable ideals and establishing invariance properties for commuting hermitian n-tuples.

Findings

Modulus of quasicentral approximation remains stable under hybrid ideal replacements.

Extension of Weyl--von Neumann theorem to hybrid normed ideals.

Use of singular integrals of mixed homogeneity in proofs.

Abstract

In hybrid normed ideal perturbations of $n$-tuples of operators, the normed ideal is allowed to vary with the component operators. We begin extending to this setting the machinery we developed for normed ideal perturbations based on the modulus of quasicentral approximation and an adaptation of our non-commutative generalization of the Weyl--von~Neumann theorem. For commuting $n$-tuples of hermitian operators, the modulus of quasicentral approximation remains essentially the same when $\cC_n^-$ is replaced by a hybrid $n$-tuple $\cC_{p_1,\dots}^-,\dots,\cC^-_{p_n}$, $p_1^{-1} + \dots + p_n^{-1} = 1$. The proof involves singular integrals of mixed homogeneity.