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

TL;DR

This paper investigates the joint spectra of tensor product representations of solvable Lie algebras on Banach spaces, providing descriptions in terms of the spectra of individual representations and extending to essential spectra and nilpotent systems.

Contribution

It introduces new descriptions of the joint spectra for tensor product representations of solvable Lie algebras, including essential spectra and applications to nilpotent operator systems.

Findings

Describes S{ }lodkowski and split joint spectra in tensor product representations.

Provides formulas for essential joint spectra in terms of component spectra.

Applies results to nilpotent and commutative operator systems.

Abstract

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 $\rho_i\colon L_i\to {\rm L}(X_i)$ of the algebras, $i=1$, $2$, we consider the Lie algebra $L=L_1\times L_2$, and the tensor product representation of $L$, $\rho\colon L\to {\rm L}(X_1\tilde{\otimes}X_2)$, $\rho=\rho_1\otimes I +I\otimes \rho_2$. 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.