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Spectra of weighted composition operators on abstract Hardy spaces. (English) Zbl 1450.47015

Summary: In the paper, the spectra of weighted composition operators on Hardy-type spaces of holomorphic functions on the unit disc of the complex plane are studied. The spectra of invertible operators induced by elliptic and parabolic automorphisms are described, for weighted composition operators acting on abstract Hardy spaces generated by Banach lattices. We also study spectra of weighted composition operators (not necessarily invertible) on Hardy-Lorentz spaces.

MSC:

47B33 Linear composition operators
47A10 Spectrum, resolvent
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46E15 Banach spaces of continuous, differentiable or analytic functions
46L70 Nonassociative selfadjoint operator algebras
30H10 Hardy spaces
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[1] Asekritova, I., Kruglyak, N.: Necessary and sufficient conditions for invertibility of operators in spaces of real interpolation. J. Funct. Anal. 264(1), 207-245 (2013) · Zbl 1279.46012 · doi:10.1016/j.jfa.2012.10.007
[2] Asekritova, I., Kruglyak, N., Mastyło, M.: Interpolation of Fredholm operators. Adv. Math. 295, 421-496 (2016) · Zbl 1364.46020 · doi:10.1016/j.aim.2016.03.009
[3] Bergh, J., Löfström, J.: Interpolation Spaces. A Series of Comprehensive Studies in Mathematics, vol. 223. Springer, Berlin (1976) · Zbl 0344.46071
[4] Bourdon, P.: Invertible weighted composition operators. Proc. Am. Math. Soc. 142, 289-299 (2013) · Zbl 1283.47027 · doi:10.1090/S0002-9939-2013-11804-6
[5] Chalendar, I., Gallardo, E., Partington, J.R.: Weighted composition operators on the Dirichlet space: boundedness and spectral properties. Math. Ann. 363(3-4), 1265-1279 (2015) · Zbl 1390.47004 · doi:10.1007/s00208-015-1195-y
[6] Cowen, C., MacCluer, B.: Composition Operators on Spaces of Analytic Functions. CRC Press, Boca Raton (1995) · Zbl 0873.47017
[7] Cowen, C.: Composition operators on \[H^2\] H2. J. Oper. Theory 9, 77-106 (1983) · Zbl 0504.47032
[8] Cowen, C., Ko, E., Thompson, D., Tian, F.: Spectra of some weighted composition operators on \[H^2\] H2. Acta Sci. Math. (Szeged) 82(1-2), 221-234 (2016) · Zbl 1399.47083 · doi:10.14232/actasm-014-542-y
[9] Eklund, T., Lindström, M., Mleczko, P.: Spectral properties of weighted composition operators on the Bloch and Dirichlet spaces. Stud. Math. 232, 95-112 (2016) · Zbl 1372.47035
[10] Grafakos, L.: Classical Fourier Analysis, III edn. Graduate Texts in Mathematics, vol. 249. Springer, New York (2014)
[11] Hyvärinen, O., Lindström, M., Nieminen, I., Saukko, E.: Spectra of weighted composition operators with automorphic symbols. J. Funct. Anal. 265, 1749-1777 (2013) · Zbl 1325.47054 · doi:10.1016/j.jfa.2013.06.003
[12] Hyvärinen, O., Nieminen, I.: Essential spectra of weighted composition operators with hyperbolic symbols. Concr. Oper. 2, 110-119 (2015) · Zbl 1321.47061
[13] Kamowitz, H.: Compact operators of the form \[u C_\varphi\] uCφ. Pac. J. Math. 80, 205-211 (1979) · Zbl 0414.47016 · doi:10.2140/pjm.1979.80.205
[14] Krein, S.G., Petunin, Yu.I., Semenov, E.M.: Interpolation of Linear Operators. Translations of Mathematical Monographs, vol. 54. American Mathematical Society, Providence (1982)
[15] Lefèvre, P., Li, D., Queffélec, H., Rodríguez-Piazza, L.: Composition operators on Hardy-Orlicz spaces. Mem. Am. Math. Soc. 207(974), vi+74 (2010) · Zbl 1200.47035
[16] Lengfield, M.: Duals and envelopes of some Hardy-Lorentz spaces. Proc. Am. Math. Soc. 133(5), 1401-1409 (2005) · Zbl 1074.32002 · doi:10.1090/S0002-9939-04-07656-7
[17] Lu, Q., Cao, G.F., Liu, L.F.: Hardy-Orlicz spaces and their multiplication operators. Acta Math. Sin. 21(3), 593-598 (2005) · Zbl 1100.46016 · doi:10.1007/s10114-005-0516-y
[18] Mastyło, M., Mleczko, P.: Norm estimates for matrix operators between Banach spaces. Linear Algebra Appl. 438(3), 986-1001 (2013) · Zbl 1304.47049 · doi:10.1016/j.laa.2012.09.002
[19] Mastyło, M., Rodríguez-Piazza, L.: Carleson measures and embeddings of abstract Hardy spaces into function lattices. J. Funct. Anal. 268, 902-928 (2015) · Zbl 1320.46024 · doi:10.1016/j.jfa.2014.11.004
[20] Xu, Q.: Notes on interpolation of Hardy spaces. Ann. Inst. Fourier 42, 875-889 (1992) · Zbl 0760.46060 · doi:10.5802/aif.1313
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