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Composition operators on analytic Lipschitz spaces. (English) Zbl 0793.47037

Let D denote the unit disk {z:|z|1} of the complex plane. If f is an analytic function on D and 0<q<1, then f q is defined by

f q =sup|f(z)-f(w)|/|z-w| q ; z D , w D,

and the space A q is defined by A q ={f:f q <}. If φ:DD is analytic, then the composition operator C φ is defined by C φ (f)(x)=fφ(x)=f(φ(x)).

The main statements of this paper indicate that

(i) C φ :A q A q is bounded if and only if

sup1-|z| 2 1-|φ(z)| 2 1-q |φ ' (z)|:|z|<1<;

and

(ii) if lim |z|1 1-|z| 2 1-|φ(z)| 2 1-q |φ ' (z)|=0, then C φ :A q A q is w-compact.

Notes indicated as added in the proof of the paper state that similar conclusions have been derived by different procedures in papers of R. C. Roan [Rocky Mt. J. Math., 10, 371-379 (1980; Zbl 0433.46023)] and J. H. Shapiro [Proc. Am. Math. Soc. 100, 49-57 (1987; Zbl 0622.47028)].


MSC:
47B38Operators on function spaces (general)
46E15Banach spaces of continuous, differentiable or analytic functions
47B07Operators defined by compactness properties
30C20Conformal mappings of special domains