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

100%

Boasso E.

Instytut Matematyczny Polskiej Akademii Nauk

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.

On the joint spectral radius of a nilpotent Lie algebra of matrices

96%

Boasso E.

Studia Mathematica

1999

tom 132

nr 1

15-27

For a complex nilpotent finite-dimensional Lie algebra of matrices, and a Jordan-Hölder basis of it, we prove a spectral radius formula which extends a well-known result for commuting matrices.

Ograniczanie wyników

1 Studia Mathematica

2 Boasso E.

1 2003

1 1999

Wyniki wyszukiwania

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

On the joint spectral radius of a nilpotent Lie algebra of matrices