×

Weighted composition operators on differentiable Lipschitz algebras. (English) Zbl 1417.46016

Summary: Let \(\operatorname{Lip}^n(X, \alpha)\) be the algebra of complex-valued functions on a perfect compact plane set \(X\), whose derivatives up to order \(n\) exist and satisfy the Lipschitz condition of order \(0<\alpha \leq 1\). We establish a necessary and sufficient condition for a weighted composition operator on \(\operatorname{Lip}^n(X, \alpha)\) to be compact. To obtain the necessary condition in the case \(0<\alpha < 1\), we provide a relation between these algebras and Zygmund-type spaces \(\mathcal {Z}_n^\alpha \). We then conclude some interesting results about weighted composition operators on \(\mathcal {Z}_n^\alpha \) and determine the spectra of these operators when they are compact or Riesz.

MSC:

46E15 Banach spaces of continuous, differentiable or analytic functions
46J15 Banach algebras of differentiable or analytic functions, \(H^p\)-spaces
47B33 Linear composition operators
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. U.S. Department of Commerce, Washington, D.C. (1964) · Zbl 0171.38503
[2] Amiri, S., Golbaharan, A., Mahyar, H.: Weighted composition operators on algebras of differentiable functions. Bull. Belg. Math. Soc. Simon-Stevin 23(4), 595-608 (2016) · Zbl 1436.47005
[3] Behrouzi, F., Mahyar, H.: Compact endomorphisms of certain analytic Lipschitz algebras. Bull. Belg. Math. Soc. Simon Stevin 12(2), 301-312 (2005) · Zbl 1110.46036
[4] Caratheodory, C.: Theory of Functions of a Complex Variable, vol. II. Chelsea, New York (1960)
[5] Dales, H.G.: Banach Algebras and Automatic Continuity, London Math. Soc. Monogr., vol. 24. Clarendon Press, Oxford (2000) · Zbl 0981.46043
[6] Dales, H.G., Davie, A.M.: Quasianalytic Banach function algebras. J. Funct. Anal. 13, 28-50 (1973) · Zbl 0254.46027 · doi:10.1016/0022-1236(73)90065-7
[7] Duren, P.L.: Theory of \[H^p\] Hp Spaces. Academic Press, San Diego (1970)
[8] Feinstein, J.F., Kamowitz, H.: Quasicompact and Riesz endomorphisms of Banach algebras. J. Funct. Anal. 225, 427-438 (2005) · Zbl 1089.46029 · doi:10.1016/j.jfa.2005.04.002
[9] Honary, T.G., Mahyar, H.: Approximation in Lipschitz algebras of infinitely differentiable functions. Bull. Korean Math. Soc. 36(4), 629-636 (1999) · Zbl 0951.46025
[10] Jarosz, \[K.: \text{ Lip }\_{Hol}(X,\alpha )\] Lip_Hol(X,α). Proc. Am. Math. Soc. 125(10), 3129-3130 (1997) · Zbl 0888.46031 · doi:10.1090/S0002-9939-97-04238-X
[11] MacCluer, B.D., Zhao, R.: Essential norms of weighted composition operators between Bloch-type spaces. Rocky Mt. J. Math. 33(4), 1437-1458 (2003) · Zbl 1061.30023 · doi:10.1216/rmjm/1181075473
[12] Mahyar, H.: Compact endomorphisms of infinitely differentiable Lipschitz algebras. Rocky Mt. J. Math. 39(1), 193-217 (2009) · Zbl 1173.46032 · doi:10.1216/RMJ-2009-39-1-193
[13] Mahyar, H., Sanatpour, A.H.: Compact composition operators on certain analytic Lipshitz spaces. Bull. Iran. Math. Soc. 38(1), 85-99 (2012) · Zbl 1320.47025
[14] Mahyar, H., Sanatpour, A.H.: Compact and quasicompact homomorphisms between differentiable Lipschitz algebras. Bull. Belg. Math. Soc. Simon Stevin 17, 485-497 (2010) · Zbl 1213.47042
[15] Ohno, S., Stroethoff, K., Zhao, R.: Weighted composition operators between Bloch-type spaces. Rocky Mt. J. Math. 33(1), 191-215 (2003) · Zbl 1042.47018 · doi:10.1216/rmjm/1181069993
[16] Zhu, K.: Spaces of Holomorphic Functions in the Unit Ball, Grad. Texts in Math., vol. 226. Springer, New York (2005) · Zbl 1067.32005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.