Joint spectra of the tensor product representation of the direct sum of two solvable Lie algebras
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 δ{turned}i: Li → L(Xi) of the algebras, i = 1;2, we consider the Lie algebra L = L1 L2 and the tensor prod...
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| Formato: | JOUR |
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| Acceso en línea: | http://hdl.handle.net/20.500.12110/paper_00123862_v_n416_p5_Boasso |
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| Sumario: | 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 δ{turned}i: Li → L(Xi) of the algebras, i = 1;2, we consider the Lie algebra L = L1 L2 and the tensor product representation of L, δ{turned}: L → L(X1 ⊕ X2), δ{turned} = δ{turned}1 I + I δ{turned}2. We study the Słodkowski and split joint spectra of the representation δ{turned}, and we describe them in terms of the corresponding joint spectra of δ{turned}1 and δ{turned}2. Moreover, we study the essential Słodkowski and essential split joint spectra of the representation δ{turned}, and we describe them by means of the corresponding joint spectra and essential joint spectra of δ{turned}1 and δ{turned}2. 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. © Instytut Matematyczny PAN, Warszawa 2003. |
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