On the spectrum of multiplication operators

TL;DR

This paper investigates the spectra of multiplication operators on B(H), relating them to spectra on Hilbert-Schmidt operators, and addresses open questions about spectral positivity and spectrum enlargement under certain conditions.

Contribution

It introduces new spectral relations for multiplication operators, solves an open problem on spectral positivity with commuting coefficients, and demonstrates spectrum enlargement phenomena.

Findings

Spectral relations between operators and their restrictions to Hilbert-Schmidt class.

Resolution of an open problem on positivity of spectra with commuting positive coefficients.

Identification of spectrum enlargement due to Wiener-Pitt phenomena under specific conditions.

Abstract

We study relations between spectra of two operators that are connected to each other through some intertwining conditions. As application we obtain new results on the spectra of multiplication operators on $B(\cl H)$ relating it to the spectra of the restriction of the operators to the ideal $\mathcal C_2$ of Hilbert-Schmidt operators. We also solve one of the problems, posed in [B.Magajna, Proc. Amer. Math. Soc, 141 2013, 1349-1360] about the positivity of the spectrum of multiplication operators with positive operator coefficients when the coefficients on one side commute. Using the Wiener-Pitt phenomena we show that the spectrum of a multiplication operator with normal coefficients satisfying the Haagerup condition might be strictly larger than the spectrum of its restriction to $\mathcal C_2$.