Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Local spectral theory and orbits of operators


Authors: T. L. Miller and V. G. Miller
Journal: Proc. Amer. Math. Soc. 127 (1999), 1029-1037
MSC (1991): Primary 47B40, 47B99
DOI: https://doi.org/10.1090/S0002-9939-99-04639-0
MathSciNet review: 1473674
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: For $T\in \mathcal{L}(X)$, we give a condition that suffices for $\varphi(T)$ to be hypercyclic where $\varphi$ is a nonconstant function that is analytic on the spectrum of $T$. In the other direction, it is shown that a property introduced by E. Bishop restricts supercyclic phenomena: if $T\in \mathcal{L}(X)$ is finitely supercyclic and has Bishop's property $(\beta)$, then the spectrum of $T$ is contained in a circle.


References [Enhancements On Off] (What's this?)

  • 1. E. Albrecht, On decomposable operators, Int. Eq. Oper. Theory 2 (1979), 1-10. MR 80h:47042
  • 2. E. Albrecht and J. Eschmeier, Analytic functional models and local spectral theory, Proc. London Math. Soc. 75 (1997), 323-348. MR 98f:47043
  • 3. E. Albrecht, J. Eschmeier and M. Neumann, Some topics in the theory of decomposable operators, Operator Theory: Adv. Appl., vol. 17, Birkhäuser, Basel, 1986, pp. 15-34. MR 89c:47046
  • 4. A. Atzmon, Power regular operators, Trans. Amer. Math. Soc. 347 (1995), 3101-3109. MR 95j:47005
  • 5. S. Ansari and P. Bourdon, Some properties of cyclic vectors, preprint.
  • 6. G. D. Birkhoff, Démonstration d'un théorem elementaire sur les fonctions entieres, C. R. Acad. Sci. Paris 189 (1929), 473-475.
  • 7. E. Bishop, A duality theory for an arbitrary operator, Pacific J. Math. 9 (1959), 379-397. MR 22:8339
  • 8. P. S. Bourdon, Orbits of hyponormal operators, Mich. Math. J. 44 (1997), 345-353. MR 98e:47037
  • 9. P. S. Bourdon and J. H. Shapiro, Cyclic Phenomena for Composition Operators, Memoirs of Amer. Math. Soc. 125 (1997), Amer. Math. Soc., Providence, RI. MR 97h:47023
  • 10. I. Colojoar\u{a} and C. Foia\c{s}, Theory of Generalized Spectral Operators, Gordon and Breach, New York, 1968. MR 52:15085
  • 11. R. G. Douglas, On extending commutative semigroups of isometries, Bull. London Math. Soc. 1 (1969), 157-159. MR 39:7458
  • 12. J. Eschmeier and B. Prunaru, Invariant subspaces for operators with Bishop's property $(\beta)$ and thick spectrum, J. Funct. Anal. 94 (1990), 196-222. MR 91m:47008
  • 13. J. Eschmeier and M. Putinar, Spectral Decomposition and Analytic Sheaves, London Math. Soc. Monographs New Series Analytic Sheaves, Oxford Univ. Press, New York, 1996. MR 98h:47002
  • 14. S. Frunz\u{a}, A complement to the duality theorem for decomposable operators, Rev. Roumaine Math. Pures Appl. 28 (1983), 475-478. MR 86h:47048
  • 15. R. M. Gethner and J. H. Shapiro, Universal vectors for operators on spaces of holomorphic functions, Proc. Amer. Math. Soc. 100 (1987), 281-288. MR 88g:47060
  • 16. G. Godefroy and J. H. Shapiro, Operators with dense, invariant, cyclic manifolds, J. Funct. Anal. 98 (1991), 229-269. MR 92d:47029
  • 17. M. A. Gold'man and S. N. Kra[??]ckovskii, Invariance of certain spaces associated with the operator $A-\lambda I$, Soviet Math. Dokl. 154 (1964), 500-502. MR 29:467
  • 18. D. Herrero, Hypercyclic operators and chaos, J. Oper. Theory 28 (1992), 93-103. MR 95g:47031
  • 19. G. Herzog and C. Schmoeger, On operators $T$ such that $f(T)$ is hypercyclic, Studia Math. 108 (1994), 209-216. MR 95f:47031
  • 20. H. M. Hilden and L. J. Wallen, Some cyclic and non-cyclic vectors of certain operators, Ind. Univ. Math. J. 23 (1974), 557-565. MR 48:4796
  • 21. T. Kato, Perturbation theory for nullity, deficiency and other quantities of linear operators, J. Anal. Math. 6 (1958), 261-322. MR 21:6541
  • 22. C. Kitai, Invariant closed sets for linear operators, Dissertation, Univ. of Toronto, 1982.
  • 23. K. B. Laursen and M. M. Neumann, An Introduction to Local Spectral Theory, London Math. Soc. Monographs New Series, Oxford Univ. Press, New York (to appear).
  • 24. K. B. Laursen and M. M. Neumann, Local spectral theory and spectral inclusions, Glasgow Math. J. 36 (1994), 331-343. MR 95k:47002
  • 25. K. B. Laursen and P. Vrbova, Some remarks on the surjectivity spectrum of linear operators, Czech. Math. J. 39 (1989), 730-739. MR 90m:47010
  • 26. J. Leiterer, Banach coherent analytic Fréchet sheaves, Math. Nachr. 85 (1978), 91-109. MR 80b:32026
  • 27. G. R. MacLane, Sequences of derivatives and normal families, J. Analyse Math. 2 (1952), 72-87. MR 14:741d
  • 28. V. G. Miller, Remarks on finitely-hypercyclic and finitely-supercyclic operators, Integral Equations Oper. Theory 29 (1997), 110-115. MR 98i:47017
  • 29. V. G. Miller and M. M. Neumann, Local spectral theory for multipliers and convolution operators, Algebraic Methods in Operator Theory (R. Curto and P. Jørgensen, eds.), Birkhäuser, Boston, 1994, pp. 25-36. MR 95m:43004
  • 30. V. Müller, On the regular spectrum, J. Oper. Theory 31 (1994), 363-380.
  • 31. C. R. Putnam, An inequality for the area of hyponormal spectra, Math. Z. 116 (1970), 323-330. MR 42:5085
  • 32. M. Putinar, Hyponormal operators are subscalar, J. Oper. Theory 12 (1984), 385-395. MR 85h:47027
  • 33. S. Rolewicz, On orbits of elements, Studia Math. 32 (1969), 17-22. MR 39:3292
  • 34. H. Salas, Hypercyclic weighted shifts, Trans. Amer. Math. Soc. 347 (1995), 993-1004. MR 95e:47042
  • 35. C. Schmoeger, Ein Spectralabbildungssatz, Arch. Math. (Basel) 55 (1990), 484-489. MR 92h:47007
  • 36. A. L. Shields, Weighted shift operators and analytic function theory, Topics in Operator Theory (C. Pearcy, ed.), Mathematical Surveys 13, Amer. Math. Soc., Providence, RI, 1974. MR 50:14341
  • 37. R. C. Smith, Local spectral theory for invertible composition operators on $H^p$, Int. Eq. and Operator Theory 25 (1996), 329-335. MR 97e:47049
  • 38. F.-H. Vasilescu, Analytic Functional Calculus and Spectral Decomposition, Editura Academiei and D. Reidel Publishing Co., Bucharest and Dordrecht, 1982. MR 85b:47016

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 47B40, 47B99

Retrieve articles in all journals with MSC (1991): 47B40, 47B99


Additional Information

T. L. Miller
Affiliation: Department of Mathematics, Mississippi State University, Drawer MA, Mississippi State, Mississippi 39762
Email: miller@math.msstate.edu

V. G. Miller
Affiliation: Department of Mathematics, Mississippi State University, Drawer MA, Mississippi State, Mississippi 39762
Email: vivien@math.msstate.edu

DOI: https://doi.org/10.1090/S0002-9939-99-04639-0
Received by editor(s): December 23, 1996
Received by editor(s) in revised form: July 11, 1997
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1999 American Mathematical Society

American Mathematical Society