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 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.
Referencias:
- Albrecht, E., On the joint spectra (1979) Studia Math, 64, pp. 263-271
- Arens, R., The analytic functional calculus in commutative topological algebras (1961) Pacific J. Math, 11, pp. 405-429
- Arens, R., Caldeŕon, P., Analytic functions of several Banach algebra elements (1955) Ann. of Math, 62, pp. 204-216
- Bernat, P., Conze, N., Duo, M., Lévy-Nahas, M., Rais, M., Renouard, M., Vergne, P., (1972) Représentations des groupes de Lie résolubles, , Dunod, Paris
- Boasso, E., Dual properties and joint spectra for solvable Lie algebras of operators (1995) J. Operator Theory, 33, pp. 105-116
- Boasso, E., Tensor products and joint spectra for solvable Lie algebras of operators (1998) Collect. Math, 49, pp. 9-16
- Boasso, E., Larotonda, A., A spectral theory for solvable Lie algebras of operators (1993) Pacific J. Math, 158, pp. 15-22
- Bourbaki, N., (1960) Éléments de mathématique. Fasc. XXVI. Groupes et algébres de Lie. Chapitre I: Algébres de Lie, , Hermann, Paris
- Ceausescu, Z., Vasilescu, F.H., Tensor products and Taylor's joint spectrum (1978) Studia Math, 62, pp. 305-311
- Ceausescu, Z., Vasilescu, F.H., Tensor products and the joint spectrum in Hilbert spaces (1978) Proc. Amer. Math. Soc, 72, pp. 505-508
- Curto, R., (1988) Applications of several complex variables to multiparameter spectral theory, pp. 25-90. , 192, in: Surveys of Some Recent Results in Operator Theory, Vol. II, J.B. Conway and B. Morrel (eds.), Pitman Res. Notes Math. Ser. Longman Sci. Tech., Harlow
- Curto, R., Fialkow, L., The spectral picture of (LA;RB) (1987) J. Funct. Anal, 71, pp. 371-392
- Eschmeier, J., (1987) Analytic spectral mapping theorems for joint spectra, pp. 167-181. , 24, in: Operators in Indefinite Metric Spaces, Scattering Theory and Other Topics (Bucharest, 1985), H. Helson et al. (eds.), Oper. Theory Adv. Appl. Birkhäuser, Basel
- Eschmeier, J., (1985) Analytic spectral mapping theorems for joint spectra, , in: Operators in Indefinite Metric Spaces, Scattering Theory and Other Topics (Bucharest
- Helson, H., Tensor products and elementary operators (1988) J. Reine Angew. Math, 390, pp. 47-66
- Faínshteín, A.S., The joint essential spectrum of a family of linear operators (1980) Funct. Anal. Appl, 14, pp. 152-153
- Faínshteín, A.S., Taylor joint spectrum for families of operators generating nilpotent Lie algebras (1993) J. Operator Theory, 29, pp. 3-27
- Hörmander, L., (1973) An Introduction to Complex Analysis in Several Variables, , North-Holland, Amsterdam
- Ichinose, T., Spectral properties of tensor products of linear operators I (1978) Trans. Amer. Math. Soc, 235, pp. 75-113
- Müller, V., The Słodkowski spectra and higher Shilov boundaries (1993) Studia Math, 105, pp. 69-75
- Ott, C., A note on a paper of E. Boasso and A. Larotonda (1996) Pacific J. Math, 173, pp. 173-179
- Ott, C., (1997) Gemeinsame Spektren auösbarer Operator-Liealgebren, , http://analysis.math.uni-kiel.de/wrobel/, Dissertation, Kielf
- Shilov, G., On the decomposition of a commutative normed ring into a direct sum of ideals (1955) Amer. Math. Soc. Transl, 1, pp. 37-48
- Słodkowski, Z., An infinite family of joint spectra (1973) Studia Math, 61, pp. 239-255
- Taylor, J.L., A joint spectrum for several commuting operators (1970) J. Funct. Anal, 6, pp. 172-191
- Vasilescu, F.H., Stability of the index of a complex of Banach spaces (1979) J. Operator Theory, 2, pp. 247-275
- Waelbroeck, L., Le calcul symbolique dans les algébres commutatives (1954) J. Math. Pures Appl, 33, pp. 147-186
- Wrobel, V., Tensor products of linear operators in Banach spaces and Taylor's joint spectrum (1986) J. Operator Theory, 16, pp. 273-283
- Wrobel, V., Tensor products of linear operators in Banach spaces and Taylor's joint spectrum. II (1988) J. Operator Theory 16 (1986), 19, pp. 3-24
Citas:
---------- APA ----------
(2003)
. Joint spectra of the tensor product representation of the direct sum of two solvable Lie algebras. Dissertationes Mathematicae(416), 5-46.
http://dx.doi.org/10.4064/dm416-0-1---------- CHICAGO ----------
Boasso, E.
"Joint spectra of the tensor product representation of the direct sum of two solvable Lie algebras"
. Dissertationes Mathematicae, no. 416
(2003) : 5-46.
http://dx.doi.org/10.4064/dm416-0-1---------- MLA ----------
Boasso, E.
"Joint spectra of the tensor product representation of the direct sum of two solvable Lie algebras"
. Dissertationes Mathematicae, no. 416, 2003, pp. 5-46.
http://dx.doi.org/10.4064/dm416-0-1---------- VANCOUVER ----------
Boasso, E. Joint spectra of the tensor product representation of the direct sum of two solvable Lie algebras. Diss. Math. 2003(416):5-46.
http://dx.doi.org/10.4064/dm416-0-1