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

Enrico Boasso. Joint spectra of the tensor product representation of the direct sum of two solvable Lie algebras. 2003. <http://eudml.org/doc/285939>.

@book{EnricoBoasso2003,
abstract = {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 $ϱ_\{i\}: L_\{i\} → L(X_\{i\})$ 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.},
author = {Enrico Boasso},
keywords = {complex Banach spaces; tensor product; multiplication representation; operator ideal; essential Słodkowski joint spectrum; essential split joint spectrum},
language = {eng},
title = {Joint spectra of the tensor product representation of the direct sum of two solvable Lie algebras},
url = {http://eudml.org/doc/285939},
year = {2003},
}

TY - BOOK
AU - Enrico Boasso
TI - Joint spectra of the tensor product representation of the direct sum of two solvable Lie algebras
PY - 2003
AB - 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 $ϱ_{i}: L_{i} → L(X_{i})$ 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.
LA - eng
KW - complex Banach spaces; tensor product; multiplication representation; operator ideal; essential Słodkowski joint spectrum; essential split joint spectrum
UR - http://eudml.org/doc/285939
ER -