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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.

MSC:

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