Tensor products and joint spectra for solvable Lie algebras of operators

TL;DR

This paper investigates the joint spectra of tensor products of solvable Lie algebras of operators on Hilbert spaces, establishing a relationship between the joint spectrum of the combined algebra and the spectra of individual algebras.

Contribution

It provides a formula expressing the joint spectrum of the tensor product of two solvable Lie algebras in terms of the Cartesian product of their individual spectra.

Findings

Joint spectrum of tensor product equals Cartesian product of individual spectra.

Results apply to solvable Lie algebras acting on Hilbert spaces.

Provides a spectral characterization for combined Lie algebra actions.

Abstract

Given two complex Hilbert spaces, $H_1$ and $H_2$, and two complex solvable finite dimensional Lie algebras of operators, $L_1$ and $L_2$, such that $L_i$ acts on $H_i$ (i= 1,2), the joint spectrum of the Lie algebra $L_1\times L_2$, which acts on $H_1\overline\otimes H_2$, is expressed by the cartesian product of $Sp(L_1,H_1)$ and $Sp(L_2,H_2)$.