On the joint spectra of the two dimensional Lie algebra of operators in Hilbert spaces

TL;DR

This paper analyzes the joint spectra of a specific two-dimensional non-commutative Lie algebra of operators on Hilbert spaces, providing reduction techniques and applying results to finite-dimensional cases.

Contribution

It introduces a method to compute joint spectra of a two-dimensional Lie algebra of operators by reducing the problem to spectra of a single operator, under certain conditions.

Findings

Reduced joint spectra computation to spectra of a single operator

Extended analysis to the case where y^2=0

Applied results to finite-dimensional Hilbert spaces

Abstract

We consider the complex solvable non-commutative two dimensional Lie algebra $L$, $L=<y>\oplus <x>$, with Lie bracket $[x,y]=y$, as linear bounded operators acting on a complex Hilbert space $H$. Under the assumption $R(y)$ closed, we reduce the computation of the joint spectra $Sp(L,E)$, $\sigma_{\delta ,k}(L,E)$ and $\sigma_{\pi ,k}(L,E)$, $k= 0,1,2$, to the computation of the spectrum, the approximate point spectrum, and the approximate compression spectrum of a single operator. Besides, we also study the case $y^2=0$, and we apply our results to the case $H$ finite dimensional.