×

On convexity of composition and multiplication operators on weighted Hardy spaces. (English) Zbl 1191.47040

Summary: A bounded linear operator \(T\) on a Hilbert space \(\mathbb H\), satisfying \(\| T^{2}h\|^2+\| h\| ^{2}\geq 2\| Th\|^2\) for every \(h\in \mathbb H\), is called a convex operator. In this paper, we give necessary and sufficient conditions under which a convex composition operator on a large class of weighted Hardy spaces is an isometry. Also, we discuss convexity of multiplication operators.

MSC:

47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
46E20 Hilbert spaces of continuous, differentiable or analytic functions
47B33 Linear composition operators
PDF BibTeX XML Cite
Full Text: DOI EuDML

References:

[1] C. C. Cowen and B. D. MacCluer, Composition Operators on Spaces of Analytic Functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, Fla, USA, 1995. · Zbl 0873.47017
[2] J. H. Shapiro, Composition Operators and Classical Function Theory, Universitext: Tracts in Mathematics, Springer, New York, NY, USA, 1993. · Zbl 0791.30033
[3] R. F. Allen and F. Colonna, “Isometries and spectra of multiplication operators on the Bloch space,” Bulletin of the Australian Mathematical Society, vol. 79, no. 1, pp. 147-160, 2009. · Zbl 1163.47027
[4] R. F. Allen and F. Colonna, “On the isometric composition operators on the Bloch space in Cn,” Journal of Mathematical Analysis and Applications, vol. 355, no. 2, pp. 675-688, 2009. · Zbl 1166.32300
[5] F. Bayart, “Similarity to an isometry of a composition operator,” Proceedings of the American Mathematical Society, vol. 131, no. 6, pp. 1789-1791, 2003. · Zbl 1055.47020
[6] B. J. Carswell and C. Hammond, “Composition operators with maximal norm on weighted Bergman spaces,” Proceedings of the American Mathematical Society, vol. 134, no. 9, pp. 2599-2605, 2006. · Zbl 1110.47016
[7] B. A. Cload, “Composition operators: hyperinvariant subspaces, quasi-normals and isometries,” Proceedings of the American Mathematical Society, vol. 127, no. 6, pp. 1697-1703, 1999. · Zbl 0917.47027
[8] F. Colonna, “Characterisation of the isometric composition operators on the Bloch space,” Bulletin of the Australian Mathematical Society, vol. 72, no. 2, pp. 283-290, 2005. · Zbl 1088.30025
[9] S. Li and S. Stević, “Weighted composition operators from H\infty to the Bloch space on the polydisc,” Abstract and Applied Analysis, vol. 2007, Article ID 48478, 13 pages, 2007. · Zbl 1152.47016
[10] S. Li and S. Stević, “Weighted composition operators from Zygmund spaces into Bloch spaces,” Applied Mathematics and Computation, vol. 206, no. 2, pp. 825-831, 2008. · Zbl 1215.47022
[11] S. Ohno and R. Zhao, “Weighted composition operators on the Bloch space,” Bulletin of the Australian Mathematical Society, vol. 63, no. 2, pp. 177-185, 2001. · Zbl 0985.47022
[12] J. H. Shapiro, “What do composition operators know about inner functions?” Monatshefte für Mathematik, vol. 130, no. 1, pp. 57-70, 2000. · Zbl 0951.47026
[13] S. Stević, “Essential norms of weighted composition operators from the \alpha -Bloch space to a weighted-type space on the unit ball,” Abstract and Applied Analysis, vol. 2008, Article ID 279691, 11 pages, 2008. · Zbl 1160.32011
[14] S. Stević, “Norms of some operators from Bergman spaces to weighted and Bloch-type spaces,” Utilitas Mathematica, vol. 76, pp. 59-64, 2008. · Zbl 1160.47027
[15] S. Stević, “Norm of weighted composition operators from Bloch space to H\mu \infty on the unit ball,” Ars Combinatoria, vol. 88, pp. 125-127, 2008. · Zbl 1224.30195
[16] S. I. Ueki and L. Luo, “Compact weighted composition operators and multiplication operaors between Hardy spaces,” Abstract and Applied Analysis, vol. 2008, Article ID 196498, 12 pages, 2008. · Zbl 1167.47020
[17] E. A. Nordgren, “Composition operators,” Canadian Journal of Mathematics, vol. 20, pp. 442-449, 1968. · Zbl 0161.34703
[18] H. Schwarz, Composition operators on Hp, Ph.D. thesis, University of Toledo, Toledo, Ohio, USA, 1969.
[19] F. Forelli, “The isometries of Hp,” Canadian Journal of Mathematics, vol. 16, pp. 721-728, 1964. · Zbl 0132.09403
[20] J. A. Cima and W. R. Wogen, “On isometries of the Bloch space,” Illinois Journal of Mathematics, vol. 24, no. 2, pp. 313-316, 1980. · Zbl 0412.47016
[21] R. J. Fleming and J. E. Jamison, Isometries on Banach Spaces: Function Spaces, vol. 129 of Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2003. · Zbl 1011.46001
[22] W. Hornor and J. E. Jamison, “Isometries of some Banach spaces of analytic functions,” Integral Equations and Operator Theory, vol. 41, no. 4, pp. 410-425, 2001. · Zbl 0995.46012
[23] M. J. Martín and D. Vukotić, “Isometries of the Bloch space among the composition operators,” Bulletin of the London Mathematical Society, vol. 39, no. 1, pp. 151-155, 2007. · Zbl 1115.47024
[24] L. J. Patton and M. E. Robbins, “Composition operators that are m-isometries,” Houston Journal of Mathematics, vol. 31, no. 1, pp. 255-266, 2005. · Zbl 1072.47022
[25] S. Stević and S. I. Ueki, “Isometries of a Bergman-Privalov-type space on the unit ball,” Discrete Dynamics in Nature and Society, vol. 2009, Article ID 725860, 16 pages, 2009. · Zbl 1178.32004
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.