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)

 
 

 

Spectral subspaces of subscalar and related operators


Authors: T. L. Miller, V. G. Miller and M. M. Neumann
Journal: Proc. Amer. Math. Soc. 132 (2004), 1483-1493
MSC (2000): Primary 47A11; Secondary 47B37, 47B40
DOI: https://doi.org/10.1090/S0002-9939-03-07217-4
Published electronically: October 3, 2003
MathSciNet review: 2053356
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: For a bounded linear operator $T\in L(X)$ on a complex Banach space $X$ and a closed subset $F$ of the complex plane $\mathbb{C},$ this note deals with algebraic representations of the corresponding analytic spectral subspace $ X_{T}(F)$ from local spectral theory. If $T$ is the restriction of a generalized scalar operator to a closed invariant subspace, then it is shown that $X_{T}(F)=E_{T}(F)=\bigcap_{\hspace{0.03cm}\lambda \notin F}\left( \lambda -T\right) ^{\hspace{0.03cm}p}X$ for all sufficiently large integers $ p,$ where $E_{T}(F)$ denotes the largest linear subspace $Y$ of $X$ for which $\left( \lambda -T\right) Y=Y$ for all $\lambda \in \mathbb{C} \setminus F.$ Moreover, for a wide class of operators $T$ that satisfy growth conditions of polynomial or Beurling type, it is shown that $X_{T}(F)$is closed and equal to $E_{T}(F).$


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

  • 1. T. Bermúdez and M. González, On the boundedness of the local resolvent function, Integral Equations Operator Theory 34 (1999), 1-8. MR 2000b:47010
  • 2. K. Clancey, Seminormal Operators, Lecture Notes in Math. 742, Springer-Verlag, New York, 1979. MR 81c:47002
  • 3. I. Colojoara and C. Foias, Theory of Generalized Spectral Operators, Gordon and Breach, New York, 1968. MR 52:15085
  • 4. P. C. Curtis and M. M. Neumann, Nonanalytic functional calculi and spectral maximal spaces, Pacific J. Math. 137 (1989), 65-85. MR 90h:47059
  • 5. J. Eschmeier and M. Putinar, Bishop's condition $(\beta )$ and rich extensions of linear operators, Indiana Univ. Math. J. 37 (1988), 325-348. MR 89k:47051
  • 6. J. Eschmeier and M. Putinar, Spectral Decompositions and Analytic Sheaves, Clarendon Press, Oxford, 1996. MR 98h:47002
  • 7. C. Foias and F.-H. Vasilescu, Nonanalytic local functional calculus, Czechoslovak Math. J. 24 (99) (1974), 270-283. MR 51:8885
  • 8. B. E. Johnson, Continuity of linear operators commuting with continuous linear operators, Trans. Amer. Math. Soc. 128 (1967), 88-102. MR 35:4748
  • 9. B. E. Johnson and A. M. Sinclair, Continuity of linear operators commuting with continuous linear operators II, Trans. Amer. Math. Soc. 146 (1969), 533-540. MR 40:4791
  • 10. K. B. Laursen and M. M. Neumann, An Introduction to Local Spectral Theory, Clarendon Press, Oxford, 2000. MR 2001k:47002
  • 11. T. L. Miller, V. G. Miller, and M. M. Neumann, Growth conditions and decomposable extensions, Contemp. Math., 321 (2003), 197-205.
  • 12. M. M. Neumann, Decomposable operators and generalized intertwining linear transformations, Operator Theory: Advances and Applications 28 (1988), 209-222. MR 89f:47049
  • 13. V. Pták and P. Vrbová, On the spectral function of a normal operator, Czechoslovak Math. J. 23 (98) (1973), 615-616. MR 49:1196
  • 14. M. Putinar, Hyponormal operators are subscalar, J. Operator Theory 12 (1984), 385-395. MR 85h:47027
  • 15. C. R. Putnam, Ranges of normal and subnormal operators, Michigan Math. J. 18 (1971), 33-36. MR 43:2550
  • 16. A. M. Sinclair, Automatic Continuity of Linear Operators, London Math. Soc. Lecture Note Series 21, Cambridge University Press, Cambridge, 1976. MR 58:7011
  • 17. P. Vrbová, Structure of maximal spectral spaces of generalized scalar operators, Czechoslovak Math. J. 23 (98) (1973), 493-496. MR 47:9338
  • 18. R. Whitley, Fuglede's commutativity theorem and $ \bigcap R(T-\lambda ),$ Canad. Math. Bull. 33 (1990), 331-334. MR 92f:47017
  • 19. J.-K. Yoo, Local spectral theory for operators on Banach spaces, Far East J. Math. Sci., Special Volume (2001), Part III, 303-311. MR 2002m:47005

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 47A11, 47B37, 47B40

Retrieve articles in all journals with MSC (2000): 47A11, 47B37, 47B40


Additional Information

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

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

M. M. Neumann
Affiliation: Department of Mathematics and Statistics, Mississippi State University, PO Drawer MA, Mississippi State, Mississippi 39762
Email: neumann@math.msstate.edu

DOI: https://doi.org/10.1090/S0002-9939-03-07217-4
Received by editor(s): August 22, 2002
Received by editor(s) in revised form: January 14, 2003
Published electronically: October 3, 2003
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2003 American Mathematical Society

American Mathematical Society