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Joint spectra of the tensor product representation of the direct sum of two solvable Lie algebras

Given two complex Banach spaces $X_1$ and $X_2$, a tensor product $X_1\tilde{\otimes} X_2$ of $X_1$ and $X_2$ in the sense of [14], two complex solvable finite dimensional Lie algebras $L_1$ and $L_2$, and two representations $ρ_i\colon L_i\to {\rm L}(X_i)$ of the algebras, $i=1$, $2$, we consider the Lie algebra $L=L_1\times L_2$, and the tensor product representation of $L$, $ρ\colon L\to {\rm L}(X_1\tilde{\otimes}X_2)$, $ρ=ρ_1\otimes I +I\otimes ρ_2$. In this work we study the Słodkowski and the split joint spectra of the representation $ρ$, and we describe them in terms of the corresponding joint spectra of $ρ_1$ and $ρ_2$. Moreover, we study the essential Słodkowski and the essential split joint spectra of the representation $ρ$, and we describe them by means of the corresponding joint spectra and the corresponding essential joint spectra of $ρ_1$ and $ρ_2$. In addition, with similar arguments we describe all the above-mentioned joint spectra for the multiplication representation in an operator ideal between Banach spaces in the sense of [14]. Finally, we consider nilpotent systems of operators, in particular commutative, and we apply our descriptions to them.