On the spectral $\nu$-continuity

TL;DR

This paper investigates the $ u$-continuity of the spectrum and its parts, providing conditions for $ u$-continuity of the approximate point spectrum and analyzing spectral continuity in specific operator classes.

Contribution

It establishes $ u$-continuity properties of the spectrum for Fredholm, essentially $G_1$, and $p$-hyponormal operators, extending spectral theory understanding.

Findings

Approximate point spectrum is upper semi-$ u$-continuous at Fredholm operators.

Sufficient conditions for $ u$-continuity of $\sigma_{ap}$ are provided.

Weyl spectrum is $ u$-continuous on essentially $G_1$ operators.

Abstract

In this paper we study the $\nu$-continuity of the spectrum and some of its parts. We show that the approximate point spectrum $\sigma_{ap}$ is upper semi-$\nu$-continuous at every Fredholm operator, then we give sufficient conditions to guarantee the $\nu$-continuity of $\sigma_{ap}$. Also we show that the restriction of the Weyl spectrum on the class of essentially $G_1$ operators is $\nu$-continuous. Finally, we investigate the $\nu$-continuity of the spectrum on the class of $p$-hyponormal operators.