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

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.