On spectra of representations and graphs. Erratum

TL;DR

This paper addresses a flaw in a previous proof related to spectra of representations and graphs, providing a corrected lemma applicable without the normality assumption and revising related theorems.

Contribution

It introduces a generalized version of a key lemma that removes the normality restriction, enabling correction of earlier results in spectral graph theory.

Findings

Corrected proof of Proposition 11

Generalized lemma applicable to non-normal operators

Clarification on application to Theorem 1.6

Abstract

Unfortunately the proof of the main result of [1], Theorem 1, has a flaw. Namely, Lemma 13 used in the proof of Proposition 11 is correct only under an additional assumption that the operator $A$ is normal (adjoint for the one-sided shift operator in $l^2(\mathbb N)$ provides a counterexample). Below we prove a version of Lemma 13 that does not require the normality assumption and apply it to prove Proposition 11. In addition, the same version of the lemma appears in paper [2] (as Lemma 3.1) where it is used in the proof of Theorem 1.6. We also explain here how to use the new version of Lemma 13 to correct the proof of Theorem 1.6 from [2].