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Cyclic phenomena for composition operators. (English) Zbl 0996.47032

Mem. Am. Math. Soc. 596, 105 p. (1997).
Composition operators (also called substitution operators or inner superposition operators) are most interesting examples of linear operators. Their theory is particularly rich in spaces of analytic functions on the unit disc, e.g., Bergman and Hardy spaces. Typically, the analytical properties (boundedness, compactness, subnormality, various spectral properties, etc.) of the composition operator \(C_\varphi: f\mapsto f\circ\varphi\) in such spaces heavily depends on the “geometrical” properties of the generating function \(\varphi\). The interested reader may find a wealth of results of this type in the monographs of R. K. Singh and J. S. Manhas [“Composition operators on function spaces” North-Holland Mathematics Studies. 179 (1993; Zbl 0788.47021)], C. C. Cowen and B. D. MacCluer [“Composition operators on spaces of analytic functions” Studies in Advanced Mathematics. (1995; Zbl 0873.47017)], or the second author [“Composition operators and classical function theory” (1993; Zbl 0791.30033)].
In the Memoir under review the authors treat another aspect of composition operators, namely their cyclicity, which means that there exists an element \(x\) in the underlying linear space \(X\) whose orbit \(\text{Orb}(C_\varphi, x):= \{x,C_\varphi x, C^2_\varphi x\), \(C^3_\varphi x,\dots\}\) has dense linear space in \(X\), or hypercyclicity, which means that even the orbit \(\text{Orb}(C_\varphi, x)\) itself is dense in \(X\). In particular, the authors characterize cyclic and hypercyclic operators \(C_\varphi\) on \(X= H^2\) first for Möbius transformations \(\varphi\), and afterwards for more general transformations. Altogether, the result is a well-written and self-contained survey on a fascinating topic on the borderline of geometric function theory, complex function spaces, and linear operator theory.

MSC:

47B33 Linear composition operators
30D55 \(H^p\)-classes (MSC2000)
47-02 Research exposition (monographs, survey articles) pertaining to operator theory
46E20 Hilbert spaces of continuous, differentiable or analytic functions
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