Hybrid Normed Ideal Perturbations of n-tuples of Operators I

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 ${\text{cC}}_n^{-}$ is replaced by a hybrid $n$-tuple ${\text{cC}}_{p_1,\dots }^{-},\dots ,{\text{cC}}_{p_n}^{-}$, $p_1^{-1}+\cdots +p_n^{-1}=1$. The proof involves singular integrals of mixed homogeneity.