## Hypercyclicity of the adjoint of weighted composition operators.(English)Zbl 1191.47008

A bounded linear operator $$T$$ on a Banach space $$X$$ is said to be hypercyclic if there exists a vector $$x\in X$$ such that the orbit $$\{ T^n x: n=0,1,2,\dots\}$$ is dense in $$T$$. Let $$X$$ be a Banach space of analytic functions on the open unit disk $$U$$.Assume that an analytic function $$\psi:U \to U$$ induces a bounded composition operator $$C_{\psi}$$ on $$X$$ and that an analytic function $$\varphi$$ on $$U$$ induces a bounded multiplication operator $$M_{\varphi}$$ on $$X$$. Let $$A=M_{\varphi}C_{\psi}$$. It is shown that, under some conditions on $$X$$, $$\psi$$, and $$\varphi$$, the adjoint operator $$A^{*}$$ cannot be hypercyclic.
The main idea in the proofs is showing that $$A$$ has an eigenvalue. Namely, it is well-known that the adjoint of an operator with an eigenvalue cannot be hypercyclic.

### MSC:

 47A16 Cyclic vectors, hypercyclic and chaotic operators 47B33 Linear composition operators
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### References:

 [1] Bourdon P S and Shapiro J H, Cyclic composition operators on H 2, Proc. Symp. Pure Math. 51(Part 2) (1990) 43–53 [2] Bourdon P S and Shapiro J H, Cyclic phenomena for composition operators, Mem. Amer. Math. Soc. 596(125) (1997) 1–150 · Zbl 0996.47032 [3] Gamelin T, Uniform Algebra (New York) (1984) [4] Godefroy G and Shapiro J H, Operators with dense invariant cyclic vector manifolds, J. Funct. Anal. 98 (1991) 229–269 · Zbl 0732.47016 · doi:10.1016/0022-1236(91)90078-J [5] Marsden J E and Hoffman M J, Basic complex analysis, 2nd ed. (New York: WH Freeman and Company) (1987) · Zbl 0644.30001 [6] Shapiro J H, Composition operators and classical function theory (New York: Springer-Verlag) (1993) · Zbl 0791.30033 [7] Shields A and Wallen L, The commutant of certain Hilbert space operators, Indian Univ. Math. J. 20 (1971) 777–788 · Zbl 0207.13801 · doi:10.1512/iumj.1971.20.20062 [8] Yousefi B and Rezaei H, Hypercyclic property of weighted composition operators, Proc. Am. Math. Soc. 135(10) (2007) 3263–3271 · Zbl 1129.47010 · doi:10.1090/S0002-9939-07-08833-8 [9] Yousefi B and Haghkhah S, Hypercyclicity of special operators on Hilbert function spaces, Czechoslovak Math. J. 57(132) (2007) 1035–1041 · Zbl 1174.47312 · doi:10.1007/s10587-007-0093-1
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