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 $X_1tilde{otimes} X_2$ 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 $rho_icolon L_ito {rm L}(X_i)$ of the algebras, $i=1$, $2$, we consider the Lie algebra $L=L_1times L_2$, and the tensor product representation of $L$, $rhocolon Lto {rm L}(X_1tilde{otimes}X_2)$, $rho=rho_1otimes I +Iotimes rho_2$. In this work we study the S{l}odkowski and the split joint spectra of the representation $rho$, and we describe them in terms of the corresponding joint spectra of $rho_1$ and $rho_2$. Moreover, we study the essential S{l}odkowski and the essential split joint spectra of the representation $rho$, and we describe them by means of the corresponding joint spectra and the corresponding essential joint spectra of $rho_1$ and $rho_2$. 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.