ABSTRACT. We consider the complex solvable non-commutative two dimensional Lie algebra L, L =< Y> Ell < 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), U6,k(L, E) and u,..,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 y2 = = 0, and we apply our results to the case H finite dimensional 1. Introduction. In  we introduced a joint spectrum for complex solvable finite dimensional Lie algebras of operators acting on a Banach space E. If L is such an algebra, and Sp(L, E) denotes its joint spectrum, Sp(L, E) is a compact non empty subset o
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