Topological structures of the sets of composition operators on the Bloch spaces. (English) Zbl 1087.47029

Given a Banach space \(X\) of analytic functions on the unit disk \(D\) of the complex plane, let \({\mathcal C}(X)\) be the set of all composition operators (induced by analytic self-maps of \(D\)) with the operator norm topology. Topological structures, especially path components, of the sets \({\mathcal C}(X)\) were barely known so far. In the case of the Hardy space \(H^2\), the problem of characterizing path components of \({\mathcal C}(H^2)\) was first posed by J. H. Shapiro and C. Sundberg [Pac. J.Math.145, 117–152 (1990; Zbl 0732.30027)] and still remains open. More recently, studies on the Bloch space \(\mathcal B\) or on the space \(H^\infty\) of bounded analytic functions have been done by several authors. In particular, C. Toews [“Topological structures on sets of composition operators” (Thesis, Univ.of Virginia) (2002); see also Integral Equations Oper.Theory 48, No. 2, 265–280 (2004; Zbl 1054.47021)] proved that compact composition operators on \(\mathcal B\) form a path connected set in \(\mathcal C(\mathcal B)\). The main result of this paper is to show the complete analogue of Toews’ result for the little Bloch space \({\mathcal B}_0\). The method of proof is different. As a consequence, the authors show that there exist two composition operators which are isolated in \({\mathcal C}(H^\infty)\) but lie in the same path component of \({\mathcal C}({\mathcal B}_0)\). Also, the authors prove that if the difference of two composition operators is compact on \({\mathcal B}_0\), then they belong to the same path component of \({\mathcal C}({\mathcal B}_0)\). Finally, the authors provide a sufficient condition for an element in \(\mathcal C(B)\) to be isolated.


47B33 Linear composition operators
46E15 Banach spaces of continuous, differentiable or analytic functions
Full Text: DOI


[1] Aleksandrov, A. B.; Anderson, J. M.; Nicolau, A., Inner functions, Bloch spaces and symmetric measures, Proc. London Math. Soc. (3), 79, 318-352 (1999) · Zbl 1085.46020
[2] Cowen, C. C.; MacCluer, B. D., Composition Operators on Spaces of Analytic Functions (1995), CRC Press: CRC Press Boca Raton, FL · Zbl 0873.47017
[3] Hosokawa, T.; Izuchi, K.; Zheng, D., Isolated points and essential components of composition operators on \(H^\infty \), Proc. Amer. Math. Soc., 130, 1765-1773 (2002) · Zbl 1008.47031
[4] T. Hosokawa, S. Ohno, Differences of composition operators on the Bloch spaces, preprint; T. Hosokawa, S. Ohno, Differences of composition operators on the Bloch spaces, preprint · Zbl 1174.47019
[5] MacCluer, B. D.; Ohno, S.; Zhao, R., Topological structure of the space of composition operators on \(H^\infty \), Integral Equations Operator Theory, 40, 481-494 (2001) · Zbl 1062.47511
[6] Madigan, K.; Matheson, A., Compact composition operators on the Bloch space, Trans. Amer. Math. Soc., 347, 2679-2687 (1995) · Zbl 0826.47023
[7] Montes-Rodríguez, A., The Pick-Schwarz lemma and composition operators on Bloch spaces, Rend. Circ. Mat. Palermo (2) Suppl., 56, 167-170 (1998) · Zbl 0927.30021
[8] Shapiro, J. H., Composition Operators and Classical Function Theory (1993), Springer-Verlag: Springer-Verlag New York · Zbl 0791.30033
[9] Shapiro, J. H.; Sundberg, C., Isolation amongst the composition operators, Pacific J. Math., 145, 117-152 (1990) · Zbl 0732.30027
[10] Smith, W., Inner functions in the hyperbolic little Bloch class, Michigan Math. J., 45, 103-114 (1998) · Zbl 0976.30018
[11] C. Toews, Topological Structures on Sets of Composition Operators, thesis, Univ. of Virginia, 2002; C. Toews, Topological Structures on Sets of Composition Operators, thesis, Univ. of Virginia, 2002
[12] Zhu, K., Operator Theory in Function Spaces (1990), Dekker: Dekker New York · Zbl 0706.47019
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.