Abstract
Let \((X, \Sigma , \mu )\) be a \(\sigma \)-finite measure space and \(W=uC_{\varphi }\) be a weighted composition operator on \(L^p(\Sigma )\) (\(1\le p<\infty \)), defined by \(W:f\mapsto u.(f\circ \varphi )\), where \(\varphi : X\rightarrow X\) is a measurable transformation and u is a weight function on X. In this paper, we study the hypercyclicity of W in terms of u, using the Radon–Nikodym derivatives and the conditional expectations. First, it is shown that if \(\varphi \) is a periodic nonsingular transformation, then W cannot be hypercyclic. The necessary conditions for the hypercyclicity of W are then given in terms of the Radon–Nikodym derivatives provided that \(\varphi \) is non-singular and finitely non-mixing. For the sufficient conditions, we also require that \(\varphi \) is normal. The weakly mixing and topologically mixing concepts are also studied for W. Moreover, under some specific conditions, we establish the subspace hypercyclicity of the adjoint operator \(W^*\) with respect to the Hilbert subspace \(L^2(\mathcal {A})\). Finally, to illustrate the results, some examples are given.
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The authors are very grateful to the anonymous referee(s) for a very careful reading of the paper and helpful comments which improved its presentation.
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Azimi, M.R., Jabbarzadeh, M.R. Hypercyclicity of Weighted Composition Operators on \(L^p\)-Spaces. Mediterr. J. Math. 19, 164 (2022). https://doi.org/10.1007/s00009-022-02086-3
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DOI: https://doi.org/10.1007/s00009-022-02086-3
Keywords
- Hypercyclic
- Orbit
- Weakly mixing
- Weighted composition
- Measurable transformation
- Normal
- Radon–Nikodym derivative
- Conditional expectation