# On the norm-closure of the class of hypercyclic operators

• Volume: 65, Issue: 2, page 157-161
• ISSN: 0066-2216

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## Abstract

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Let T be a bounded linear operator acting on a complex, separable, infinite-dimensional Hilbert space and let f: D → ℂ be an analytic function defined on an open set D ⊆ ℂ which contains the spectrum of T. If T is the limit of hypercyclic operators and if f is nonconstant on every connected component of D, then f(T) is the limit of hypercyclic operators if and only if $f\left({\sigma }_{W}\left(T\right)\right)\cup z\in ℂ:|z|=1$ is connected, where ${\sigma }_{W}\left(T\right)$ denotes the Weyl spectrum of T.

## How to cite

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Christoph Schmoeger. "On the norm-closure of the class of hypercyclic operators." Annales Polonici Mathematici 65.2 (1997): 157-161. <http://eudml.org/doc/269977>.

@article{ChristophSchmoeger1997,
abstract = {Let T be a bounded linear operator acting on a complex, separable, infinite-dimensional Hilbert space and let f: D → ℂ be an analytic function defined on an open set D ⊆ ℂ which contains the spectrum of T. If T is the limit of hypercyclic operators and if f is nonconstant on every connected component of D, then f(T) is the limit of hypercyclic operators if and only if $f(σ_\{W\}(T)) ∪ \{z ∈ ℂ: |z| = 1\}$ is connected, where $σ_\{W\}(T)$ denotes the Weyl spectrum of T.},
author = {Christoph Schmoeger},
journal = {Annales Polonici Mathematici},
keywords = {hypercyclic operators; Weyl spectrum},
language = {eng},
number = {2},
pages = {157-161},
title = {On the norm-closure of the class of hypercyclic operators},
url = {http://eudml.org/doc/269977},
volume = {65},
year = {1997},
}

TY - JOUR
AU - Christoph Schmoeger
TI - On the norm-closure of the class of hypercyclic operators
JO - Annales Polonici Mathematici
PY - 1997
VL - 65
IS - 2
SP - 157
EP - 161
AB - Let T be a bounded linear operator acting on a complex, separable, infinite-dimensional Hilbert space and let f: D → ℂ be an analytic function defined on an open set D ⊆ ℂ which contains the spectrum of T. If T is the limit of hypercyclic operators and if f is nonconstant on every connected component of D, then f(T) is the limit of hypercyclic operators if and only if $f(σ_{W}(T)) ∪ {z ∈ ℂ: |z| = 1}$ is connected, where $σ_{W}(T)$ denotes the Weyl spectrum of T.
LA - eng
KW - hypercyclic operators; Weyl spectrum
UR - http://eudml.org/doc/269977
ER -

## References

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1. [1] C. Bosch, C. Hernández, E. De Oteyza and C. Pearcy, Spectral pictures of functions of operators, J. Operator Theory 8 (1982), 391-400. Zbl0497.47002
2. [2] B. Gramsch and D. Lay, Spectral mapping theorems for essential spectra, Math. Ann. 192 (1971), 17-32. Zbl0203.45601
3. [3] D. A. Herrero, Limits of hypercyclic and supercyclic operators, J. Funct. Anal. 99 (1991), 179-190. Zbl0758.47016
4. [4] G. Herzog and C. Schmoeger, On operators T such that f(T) is hypercyclic, Studia Math. 108 (1994), 209-216. Zbl0818.47011
5. [5] H. Heuser, Funktionalanalysis, 2nd ed., Teubner, Stuttgart, 1986.
6. [6] K. K. Oberai, Spectral mapping theorem for essential spectra, Rev. Roumaine Math. Pures Appl. 25 (1980), 365-373. Zbl0439.47008
7. [7] C. Pearcy, Some Recent Developments in Operator Theory, CBMS Regional Conf. Ser. in Math. 36, Amer. Math. Soc., Providence, 1978.
8. [8] C. Schmoeger, Ascent, descent and the Atkinson region in Banach algebras, II, Ricerche Mat. 42 (1993), 249-264. Zbl0807.46054

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