Joint spectrum of subnormal 𝑛-tuples of composition operators

Giménez, José

doi:10.1090/S0002-9939-01-06304-3

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ISSN 1088-6826(online) ISSN 0002-9939(print)

Joint spectrum of subnormal -tuples of composition operators

Author: José Giménez
Journal: Proc. Amer. Math. Soc. 130 (2002), 2015-2023
MSC (2000): Primary 47B33; Secondary 47A13, 47B20
Posted: December 27, 2001
MathSciNet review: 1896036
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Abstract: We compute the joint (Taylor) spectrum of an $\, n$ -tuple of commuting composition operators acting on the Hardy space $\, \mathit{H}^2. \,$

1. Hari Bercovici, Operator theory and arithmetic in 𝐻^{∞}, Mathematical Surveys and Monographs, vol. 26, American Mathematical Society, Providence, RI, 1988. MR 954383 (90e:47001)
2. Earl Berkson and Horacio Porta, Semigroups of analytic functions and composition operators, Michigan Math. J. 25 (1978), no. 1, 101–115. MR 0480965 (58 #1112)
3. Carl C. Cowen, Composition operators on 𝐻², J. Operator Theory 9 (1983), no. 1, 77–106. MR 695941 (84d:47038)
4. Carl C. Cowen, Linear fractional composition operators on 𝐻², Integral Equations Operator Theory 11 (1988), no. 2, 151–160. MR 928479 (89b:47044), http://dx.doi.org/10.1007/BF01272115
5. Carl C. Cowen and Barbara D. MacCluer, Composition operators on spaces of analytic functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1995. MR 1397026 (97i:47056)
6. Carl C. Cowen and Thomas L. Kriete III, Subnormality and composition operators on 𝐻², J. Funct. Anal. 81 (1988), no. 2, 298–319. MR 971882 (90c:47055), http://dx.doi.org/10.1016/0022-1236(88)90102-4
7. Raúl E. Curto, Applications of several complex variables to multiparameter spectral theory, Surveys of some recent results in operator theory, Vol. II, Pitman Res. Notes Math. Ser., vol. 192, Longman Sci. Tech., Harlow, 1988, pp. 25–90. MR 976843 (90d:47007)
8. Raúl E. Curto and Lawrence A. Fialkow, Elementary operators with 𝐻^{∞}-symbols, Integral Equations Operator Theory 10 (1987), no. 5, 707–720. Toeplitz lectures 1987 (Tel-Aviv, 1987). MR 904485 (88j:47016), http://dx.doi.org/10.1007/BF01195797
9. Raúl E. Curto, Paul S. Muhly, and Jingbo Xia, Hyponormal pairs of commuting operators, Contributions to operator theory and its applications (Mesa, AZ, 1987), Oper. Theory Adv. Appl., vol. 35, Birkhäuser, Basel, 1988, pp. 1–22. MR 1017663 (90m:47037)
10. Farhad Jafari, Barbara D. MacCluer, Carl C. Cowen, and A. Duane Porter (eds.), Studies on composition operators, Contemporary Mathematics, vol. 213, American Mathematical Society, Providence, RI, 1998. MR 1601044 (98h:47005)
11. C. Foiaş and W. Mlak, The extended spectrum of completely non-unitary contractions and the spectral mapping theorem, Studia Math. 26 (1966), 239–245. MR 0200722 (34 #610)
12. J. Giménez, Joint hyponormality of composition operators with linear fractional symbols, Preprint, 2000.
13. Takasi Itô, On the commutative family of subnormal operators, J. Fac. Sci. Hokkaido Univ. Ser. I 14 (1958), 1–15. MR 0107177 (21 #5902)
14. Klaus-Jochen Engel and Rainer Nagel, One-parameter semigroups for linear evolution equations, Graduate Texts in Mathematics, vol. 194, Springer-Verlag, New York, 2000. With contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt. MR 1721989 (2000i:47075)
15. Abram I. Plesner, Spectral theory of linear operators. Vols. I, II, Translated from the Russian by Merlynd K. Nestell and Alan G. Gibbs, Frederick Ungar Publishing Co., New York, 1969. MR 0244792 (39 #6106)
16. Joel H. Shapiro, Composition operators and classical function theory, Universitext: Tracts in Mathematics, Springer-Verlag, New York, 1993. MR 1237406 (94k:47049)
17. Aristomenis G. Siskakis, Semigroups of composition operators on spaces of analytic functions, a review, Studies on composition operators (Laramie, WY, 1996) Contemp. Math., vol. 213, Amer. Math. Soc., Providence, RI, 1998, pp. 229–252. MR 1601120 (98m:47049), http://dx.doi.org/10.1090/conm/213/02862
18. Béla Sz.-Nagy, Spektraldarstellung linearer Transformationen des Hilbertschen Raumes, Berichtigter Nachdruck. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 39, Springer-Verlag, Berlin, 1967 (German). MR 0213890 (35 #4744)
19. Béla Sz.-Nagy and Ciprian Foiaș, Harmonic analysis of operators on Hilbert space, Translated from the French and revised, North-Holland Publishing Co., Amsterdam, 1970. MR 0275190 (43 #947)
20. Joseph L. Taylor, A joint spectrum for several commuting operators, J. Functional Analysis 6 (1970), 172–191. MR 0268706 (42 #3603)
21. Joseph L. Taylor, The analytic-functional calculus for several commuting operators, Acta Math. 125 (1970), 1–38. MR 0271741 (42 #6622)

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