Multiples of hypercyclic operators

Badea, Catalin; Grivaux, Sophie; Müller, Vladimir

doi:10.1090/S0002-9939-08-09696-2

Multiples of hypercyclic operators
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by Catalin Badea, Sophie Grivaux and Vladimir Müller PDF

Proc. Amer. Math. Soc. 137 (2009), 1397-1403 Request permission

Abstract:

We give a negative answer to a question of Prăjitură by showing that there exists an invertible bilateral weighted shift $T$ on $\ell _2(\mathbb {Z})$ such that $T$ and $3T$ are hypercyclic but $2T$ is not. Moreover, any $G_\delta$ set $M \subseteq (0,\infty )$ which is bounded and bounded away from zero can be realized as $M=\{t>0 \mid tT \textrm { is hypercyclic}\}$ for some invertible operator $T$ acting on a Hilbert space.

References

F. Bayart, É. Matheron, Dynamics of Linear Operators, Cambridge University Press, to appear.
Juan Bès and Alfredo Peris, Hereditarily hypercyclic operators, J. Funct. Anal. 167 (1999), no. 1, 94–112. MR 1710637, DOI 10.1006/jfan.1999.3437
M. De La Rosa, C. Read, A hypercyclic operator whose direct sum $T\oplus T$ is not hypercyclic, to appear in J. Operator Theory.
Fernando León-Saavedra and Vladimír Müller, Rotations of hypercyclic and supercyclic operators, Integral Equations Operator Theory 50 (2004), no. 3, 385–391. MR 2104261, DOI 10.1007/s00020-003-1299-8
V. Müller, Three problems, Oberwolfach Reports 37 (2006), p. 2272.
G. Prăjitură, personal communication.
Héctor N. Salas, Hypercyclic weighted shifts, Trans. Amer. Math. Soc. 347 (1995), no. 3, 993–1004. MR 1249890, DOI 10.1090/S0002-9947-1995-1249890-6