Given two complex Banach spaces X₁ and X₂, a tensor product X₁ ⊗̃ X₂ of X₁ and X₂ in the sense of [14], two complex solvable finite-dimensional Lie algebras L₁ and L₂, and two representations of the algebras, i = 1,2, we consider the Lie algebra L = L₁ × L₂ and the tensor product representation of L, ϱ: L → L(X₁ ⊗̃ X₂), ϱ = ϱ₁ ⊗ I + I ⊗ ϱ₂. We study the Słodkowski and split joint spectra of the representation ϱ, and we describe them in terms of the corresponding joint spectra of ϱ₁ and ϱ₂. Moreover, we study the essential Słodkowski and essential split joint spectra of the representation ϱ, and we describe them by means of the corresponding joint spectra and essential joint spectra of ϱ₁ and ϱ₂. 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.
@book{bwmeta1.element.bwnjournal-rm-doi-10_4064-dm416-0-1,
author = {Enrico Boasso},
title = {Joint spectra of the tensor product representation of the direct sum of two solvable Lie algebras},
series = {GDML\_Books},
year = {2003},
zbl = {1033.47003},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-rm-doi-10_4064-dm416-0-1}
}
Enrico Boasso. Joint spectra of the tensor product representation of the direct sum of two solvable Lie algebras. GDML_Books (2003), http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-rm-doi-10_4064-dm416-0-1/