### On the spectral set of a solvable Lie algebra of operators.

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Given two complex Banach spaces X₁ and X₂, a tensor product X₁ ⊗̃ X₂ of X₁ and X₂ in the sense of [14], two complex solvable finite-dimensional Lie algebras L₁ and L₂, and two representations ${\varrho}_{i}:{L}_{i}\to L\left({X}_{i}\right)$ of the algebras, i = 1,2, we consider the Lie algebra L = L₁ × L₂ and the tensor product representation of L, ϱ: L → L(X₁ ⊗̃ X₂), ϱ = ϱ₁ ⊗ I + I ⊗ ϱ₂. We study the Słodkowski and split joint spectra of the representation ϱ, and we describe them in terms of the corresponding joint spectra of ϱ₁ and ϱ₂. Moreover,...

For a complex nilpotent finite-dimensional Lie algebra of matrices, and a Jordan-Hölder basis of it, we prove a spectral radius formula which extends a well-known result for commuting matrices.

In this article, two results regarding the Moore-Penrose inverse in the frame of C*-algebras are considered. In first place, a characterization of the so-called reverse order law is given, which provides a solution of a problem posed by M. Mbekhta. On the other hand, Moore-Penrose hermitian elements, that is C*-algebra elements which coincide with their Moore-Penrose inverse, are introduced and studied. In fact, these elements will be fully characterized both in the Hilbert space and in the C*-algebra...

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