Quasisimilarity and compact perturbations

TL;DR

This work addresses multi-variable Fredholm theory for commuting $n$-tuples by focusing on tensor-product operators and quasisimilarity, motivated by known failures of the perturbation property in several variables. It develops a framework using Bishop's property ($\beta$), SVEP, and Taylor-type spectra to show invariance of $\sigma$, $\sigma_e$, and $\mathrm{ind}$ under quasisimilarity for tensor-product $n$-tuples. It then proves a positive perturbation result: Weyl but non-invertible tensor-product $n$-tuples formed from certain operator classes admit a compact perturbation that yields an invertible tuple. Corollaries extend the result to class $A$ operators, advancing multi-variable spectral stability and offering a constructive approach to invertibility via compact perturbations. These contributions enhance the understanding of tensor-product operator systems and their spectral properties in the Fredholm theory context.

Abstract

In this paper we show that quasisimilar $n$-tuples of tensor products of $m$-isometric operators have the same spectra, essential spectra and indices. The properties of single Fredholm operators possess \cite{4} is related to an important property which has a leading role on the theory of Fredholm operators: Fredholm n-tuples of operators. It is well known that a Fredholm operator of index zero can be perturbed by a compact operator to an invertible operator. In \cite[Problem 3]{5} the author asked if this property holds in several variables. R. Gelca in \cite{10} gave an example showing that this perturbation property fails in several variables. In this paper we give a positive answer to this question in case of tensor products of some classes of operators.

Theorems & Definitions (18)

• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• proof
• Theorem 2.5
• proof
• Lemma 2.6
• proof
• Corollary 2.7