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Hypercyclic property of weighted composition operators. (English) Zbl 1129.47010
Let \(U\) be the open unit disk and let \(\varphi\) be an holomorphic self map. The study of when the composition operators \(C_{\varphi}(f)=C_{\varphi}\circ f\) on the Hardy space \(H^2\) are hypercyclic (i.e, there is \(g \in H^2\) with \(\{C_{\varphi}^n(f):n \in \mathbb N\}\) dense in \(H^2\)) was initiated by P. S. Bourdon and J. H. Shapiro [Mem. Am. Math. Soc. 596 (1997; Zbl 0996.47032)]. Other authors have studied the same and related questions on more general spaces, for instance, E. A. Gallardo–Gutiérrez and A. Montes–Rodríguez [Mem. Am. Math. Soc. 791 (2004; Zbl 1054.47008)]. J. H. Shapiro showed in [“Lectures on Composition Operators and Analytic Function Theory” (Lecture notes from a course given at the University of Padua during the Summer of 1998, available from the author’s homepage at http://www.math.msu.edu/~shapiro/Pubvit/Downloads/CompIntro/CompIntro.html] that \(C_{\varphi}\) is hypercyclic on \(H(U)=\{\text{holomorphic functions on U}\}\) if and only if \(\varphi\) is univalent and does not have a fixed point on \(U\). Moreover, he showed that these operators are chaotic which means that they, in addition, have a dense set of periodic points.
The present authors study hypercyclicity for weighted composition operators \(M_{f}C_{\varphi}\) on \(H(U),\) where \(f \in H(U)\) and \(\varphi\) is a self map on \(U.\) They show several interesting results, among them that when \(| \lambda| =1\) and \(C_{\varphi}\) is hypercyclic, then so is \(\lambda C_{\varphi}.\) They point out that this is not contained in the result by F. León–Saavedra and V. Müller [Integral Equations Oper. Theory 50, No. 3, 385–391 (2004; Zbl 1079.47013)] which is for Banach space operators. The authors give also some necessary and sufficient conditions for the adjoint of weighted composition operators for being hypercyclic on Hilbert spaces of analytic functions. In this regard, see also C. C. Cowen and E. A. Gallardo–Gutiérrez [J. Funct. Anal. 238, No. 2, 447–462 (2006; Zbl 1106.47023)].

MSC:
47A16 Cyclic vectors, hypercyclic and chaotic operators
47B33 Linear composition operators
47B38 Linear operators on function spaces (general)
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