On the joint spectra of the two dimensional Lie algebra of operators in 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.

7 pages, original research article

References in corpus (1)

• Dual properties and joint spectra for solvable Lie algebras of operators