## 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

### Keywords:

composition operator; Bloch space; little Bloch space

### Citations:

Zbl 0732.30027; Zbl 1054.47021
Full Text:

### References:

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