Differences of composition operators on the space of bounded analytic functions in the polydisc.(English)Zbl 1160.32009

Summary: This paper gives some estimates of the essential norm for the difference of composition operators induced by $$\varphi$$ and $$\psi$$ acting on the space $$H^{\infty }(D^{n})$$ of bounded analytic functions on the unit polydisc $$D^{n}$$, where $$\varphi$$ and $$\psi$$ are holomorphic self-maps of $$D^{n}$$. As a consequence, one obtains conditions in terms of the Carathéodory distance on $$D^{n}$$ that characterizes those pairs of holomorphic self-maps of the polydisc for which the difference of two composition operators on $$H^{\infty }(D^{n})$$ is compact.

MSC:

 32A37 Other spaces of holomorphic functions of several complex variables (e.g., bounded mean oscillation (BMOA), vanishing mean oscillation (VMOA)) 47B33 Linear composition operators
Full Text:

References:

 [1] D. D. Clahane and S. Stević, “Norm equivalence and composition operators between Bloch/Lipschitz spaces of the ball,” Journal of Inequalities and Applications, vol. 2006, Article ID 61018, 11 pages, 2006. · Zbl 1131.47018 [2] 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 [3] P. Gorkin and B. D. MacCluer, “Essential norms of composition operators,” Integral Equations and Operator Theory, vol. 48, no. 1, pp. 27-40, 2004. · Zbl 1065.47027 [4] S. Li and S. Stević, “Composition followed by differentiation between Bloch type spaces,” Journal of Computational Analysis and Applications, vol. 9, no. 2, pp. 195-205, 2007. · Zbl 1132.47026 [5] S. Li and S. Stević, “Weighted composition operators from \alpha -Bloch space to H\infty on the polydisc,” Numerical Functional Analysis and Optimization, vol. 28, no. 7-8, pp. 911-925, 2007. · Zbl 1130.47015 [6] 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 1130.47015 [7] S. Li and S. Stević, “Generalized composition operators on Zygmund spaces and Bloch type spaces,” Journal of Mathematical Analysis and Applications, vol. 338, no. 2, pp. 1282-1295, 2008. · Zbl 1135.47021 [8] S. Li and H. Wulan, “Composition operators on QK spaces,” Journal of Mathematical Analysis and Applications, vol. 327, no. 2, pp. 948-958, 2007. · Zbl 1140.47306 [9] B. D. MacCluer and R. Zhao, “Essential norms of weighted composition operators between Bloch-type spaces,” The Rocky Mountain Journal of Mathematics, vol. 33, no. 4, pp. 1437-1458, 2003. · Zbl 1061.30023 [10] A. Montes-Rodríguez, “Weighted composition operators on weighted Banach spaces of analytic functions,” Journal of the London Mathematical Society, vol. 61, no. 3, pp. 872-884, 2000. · Zbl 0959.47016 [11] S. Ohno, K. Stroethoff, and R. Zhao, “Weighted composition operators between Bloch-type spaces,” The Rocky Mountain Journal of Mathematics, vol. 33, no. 1, pp. 191-215, 2003. · Zbl 1042.47018 [12] J. H. Shapiro, Composition Operators and Classical Function Theory, Universitext: Tracts in Mathematics, Springer, New York, NY, USA, 1993. · Zbl 0791.30033 [13] J. H. Shapiro, “The essential norm of a composition operator,” The Annals of Mathematics, vol. 125, no. 2, pp. 375-404, 1987. · Zbl 0642.47027 [14] J. H. Shapiro, “Compact composition operators on spaces of boundary-regular holomorphic functions,” Proceedings of the American Mathematical Society, vol. 100, no. 1, pp. 49-57, 1987. · Zbl 0622.47028 [15] J. H. Shapiro and C. Sundberg, “Isolation amongst the composition operators,” Pacific Journal of Mathematics, vol. 145, no. 1, pp. 117-152, 1990. · Zbl 0732.30027 [16] J. Shi and L. Luo, “Composition operators on the Bloch space of several complex variables,” Acta Mathematica Sinica, vol. 16, no. 1, pp. 85-98, 2000. · Zbl 0967.32007 [17] S. Stević, “Composition operators between H\infty and \alpha -Bloch spaces on the polydisc,” Zeitschrift für Analysis und ihre Anwendungen, vol. 25, no. 4, pp. 457-466, 2006. · Zbl 1118.47015 [18] S. Stević, “Weighted composition operators between mixed norm spaces and H\alpha \infty spaces in the unit ball,” Journal of Inequalities and Applications, vol. 2007, Article ID 28629, 9 pages, 2007. · Zbl 1138.47019 [19] S.-I. Ueki and L. Luo, “Compact weighted composition operators and multiplication operators between Hardy spaces,” Abstract and Applied Analysis, vol. 2008, Article ID 196498, 12 pages, 2008. · Zbl 1167.47020 [20] Z.-H. Zhou and R.-Y. Chen, “Weighted composition operators from F(p,q,s) to Bloch type spaces on the unit ball,” International Journal of Mathematics, vol. 19, no. 8, pp. 899-926, 2008. · Zbl 1163.47021 [21] Z. Zhou and Y. Liu, “The essential norms of composition operators between generalized Bloch spaces in the polydisc and their applications,” Journal of Inequalities and Applications, vol. 2006, Article ID 90742, 22 pages, 2006. · Zbl 1131.47021 [22] Z. Zhou and J. Shi, “Composition operators on the Bloch space in polydiscs,” Complex Variables, vol. 46, no. 1, pp. 73-88, 2001. · Zbl 1026.47018 [23] Z. Zhou and J. Shi, “Compactness of composition operators on the Bloch space in classical bounded symmetric domains,” The Michigan Mathematical Journal, vol. 50, no. 2, pp. 381-405, 2002. · Zbl 1044.47021 [24] E. Berkson, “Composition operators isolated in the uniform operator topology,” Proceedings of the American Mathematical Society, vol. 81, no. 2, pp. 230-232, 1981. · Zbl 0464.30027 [25] B. MacCluer, S. Ohno, and R. Zhao, “Topological structure of the space of composition operators on H\infty ,” Integral Equations and Operator Theory, vol. 40, no. 4, pp. 481-494, 2001. · Zbl 1062.47511 [26] T. Hosokawa, K. Izuchi, and D. Zheng, “Isolated points and essential components of composition operators on H\infty ,” Proceedings of the American Mathematical Society, vol. 130, no. 6, pp. 1765-1773, 2002. · Zbl 1008.47031 [27] P. Gorkin, R. Mortini, and D. Suárez, “Homotopic composition operators on H\infty (BN),” in Function Spaces (Edwardsville, IL, 2002), vol. 328 of Contemporary Mathematics, pp. 177-188, American Mathematical Society, Providence, RI, USA, 2003. · Zbl 1060.47031 [28] C. Toews, “Topological components of the set of composition operators on H\infty (BN),” Integral Equations and Operator Theory, vol. 48, no. 2, pp. 265-280, 2004. · Zbl 1054.47021 [29] J. Moorhouse, “Compact differences of composition operators,” Journal of Functional Analysis, vol. 219, no. 1, pp. 70-92, 2005. · Zbl 1087.47032 [30] T. Hosokawa and S. Ohno, “Topological structures of the sets of composition operators on the Bloch spaces,” Journal of Mathematical Analysis and Applications, vol. 314, no. 2, pp. 736-748, 2006. · Zbl 1087.47029 [31] T. Hosokawa and S. Ohno, “Differences of composition operators on the Bloch spaces,” Journal of Operator Theory, vol. 57, no. 2, pp. 229-242, 2007. · Zbl 1174.47019 [32] H. S. Bear, Lectures on Gleason Parts, vol. 121 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1970. · Zbl 0203.44601 [33] M. Jarnicki and P. Pflug, Invariant Distances and Metrics in Complex Analysis, vol. 9 of de Gruyter Expositions in Mathematics, Walter de Gruyter, Berlin, Germany, 1993. · Zbl 0789.32001 [34] J. B. Garnett, Bounded Analytic Functions, vol. 96 of Pure and Applied Mathematics, Academic Press, New York, NY, USA, 1981. · Zbl 0469.30024 [35] R. Mortini and R. Rupp, “Sums of holomorphic selfmaps of the unit disk,” Annales Universitatis Mariae Curie-Skłodowska. Sectio A, vol. 61, pp. 107-115, 2007. · Zbl 1146.30006
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.