Joint spectra of the tensor product representation of the direct sum of two solvable Lie algebras
Tom 416 / 2003
Streszczenie
Given two complex Banach spaces $X_1$ and $X_2$, a tensor product $X_1\mathbin{\widetilde{\otimes}} X_2$ 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 $\varrho_i\colon\, L_i\to {\rm L}(X_i)$ of the algebras, $i=1,2$, we consider the Lie algebra $L=L_1\times L_2$ and the tensor product representation of $L$, $\varrho\colon\, L\to {\rm L}(X_1\mathbin{\widetilde{\otimes}}X_2)$, $\varrho=\varrho_1\otimes I +I\otimes \varrho_2$. We study the Słodkowski and split joint spectra of the representation $\varrho$, and we describe them in terms of the corresponding joint spectra of $\varrho_1$ and $\varrho_2$. Moreover, we study the essential Słodkowski and essential split joint spectra of the representation $\varrho$, and we describe them by means of the corresponding joint spectra and essential joint spectra of $\varrho_1$ and $\varrho_2$. In addition, using 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.
