A hypercyclic operator whose adjoint is also hypercyclic

Author:
Héctor Salas

Journal:
Proc. Amer. Math. Soc. **112** (1991), 765-770

MSC:
Primary 47A65; Secondary 47B37

MathSciNet review:
1049848

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Abstract | References | Similar Articles | Additional Information

Abstract: An operator acting on a Hilbert space is hypercyclic if, for some vector in , the orbit is dense in . We show the existence of a hypercyclic operator--in fact, a bilateral weighted shift--whose adjoint is also hypercyclic. This answers positively a question of D. A. Herrero.

**[1]**Constantin Apostol, Lawrence A. Fialkow, Domingo A. Herrero, and Dan Voiculescu,*Approximation of Hilbert space operators. Vol. II*, Research Notes in Mathematics, vol. 102, Pitman (Advanced Publishing Program), Boston, MA, 1984. MR**735080****[2]**Robert M. Gethner and Joel H. Shapiro,*Universal vectors for operators on spaces of holomorphic functions*, Proc. Amer. Math. Soc.**100**(1987), no. 2, 281–288. MR**884467**, 10.1090/S0002-9939-1987-0884467-4**[3]**Gilles Godefroy and Joel H. Shapiro,*Operators with dense, invariant, cyclic vector manifolds*, J. Funct. Anal.**98**(1991), no. 2, 229–269. MR**1111569**, 10.1016/0022-1236(91)90078-J**[4]**Israel Halperin, Carol Kitai, and Peter Rosenthal,*On orbits of linear operators*, J. London Math. Soc. (2)**31**(1985), no. 3, 561–565. MR**812786**, 10.1112/jlms/s2-31.3.561**[5]**Domingo A. Herrero,*The diagonal entries in the formula “quasitriangular - compact = triangular” and restrictions of quasitriangularity*, Trans. Amer. Math. Soc.**298**(1986), no. 1, 1–42. MR**857432**, 10.1090/S0002-9947-1986-0857432-4**[6]**Domingo A. Herrero,*Spectral pictures of operators in the Cowen-Douglas class ℬ_{𝓃}(Ω) and its closure*, J. Operator Theory**18**(1987), no. 2, 213–222. MR**915506****[7]**Xiao Hong Cao and Yong Ge Wang,*Hypercyclic and supercyclic operators*, Qufu Shifan Daxue Xuebao Ziran Kexue Ban**24**(1998), no. 4, 4–7 (Chinese, with English and Chinese summaries). MR**1670822****[8]**Domingo A. Herrero and Warren R. Wogen,*On the multiplicity of 𝑇⊕𝑇⊕\cdots⊕𝑇*, Proceedings of the Seventh Great Plains Operator Theory Seminar (Lawrence, KS, 1987), 1990, pp. 445–466. MR**1065843**, 10.1216/rmjm/1181073120**[9]**Domingo A. Herrero and Zong Yao Wang,*Compact perturbations of hypercyclic and supercyclic operators*, Indiana Univ. Math. J.**39**(1990), no. 3, 819–829. MR**1078739**, 10.1512/iumj.1990.39.39039**[10]**C. Kitai,*Invariant closed sets for linear operators*, Thesis, Univ. of Toronto, 1982.

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Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9939-1991-1049848-8

Keywords:
Cyclic vectors,
hypercyclic vectors and operators,
weighted shifts

Article copyright:
© Copyright 1991
American Mathematical Society