Spectral localization, power boundedness and invariant subspaces under Ritt's type condition

Yu. Lyubich

Studia Mathematica (1999)

  • Volume: 134, Issue: 2, page 153-167
  • ISSN: 0039-3223

Abstract

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For a bounded linear operator T in a Banach space the Ritt resolvent condition ∥ R λ ( T ) ∥ ≤ C / | λ - 1 | (|λ| > 1) can be extended (changing the constant C) to any sector |arg(λ - 1)| ≤ π - δ, a r c c o s ( C - 1 ) < δ < π / 2 . This implies the power boundedness of the operator T. A key result is that the spectrum σ(T) is contained in a special convex closed domain. A generalized Ritt condition leads to a similar localization result and then to a theorem on invariant subspaces.

How to cite

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Lyubich, Yu.. "Spectral localization, power boundedness and invariant subspaces under Ritt's type condition." Studia Mathematica 134.2 (1999): 153-167. <http://eudml.org/doc/216629>.

@article{Lyubich1999,
author = {Lyubich, Yu.},
journal = {Studia Mathematica},
keywords = {power bounded operator; Nevanlinna's theorem; Katznelson-Tzafriri theorem},
language = {eng},
number = {2},
pages = {153-167},
title = {Spectral localization, power boundedness and invariant subspaces under Ritt's type condition},
url = {http://eudml.org/doc/216629},
volume = {134},
year = {1999},
}

TY - JOUR
AU - Lyubich, Yu.
TI - Spectral localization, power boundedness and invariant subspaces under Ritt's type condition
JO - Studia Mathematica
PY - 1999
VL - 134
IS - 2
SP - 153
EP - 167
LA - eng
KW - power bounded operator; Nevanlinna's theorem; Katznelson-Tzafriri theorem
UR - http://eudml.org/doc/216629
ER -

References

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  1. [1] K. Hoffman, Banach Spaces of Analytic Functions, Prentice-Hall, 1962. Zbl0117.34001
  2. [2] Y. Katznelson and L. Tzafriri, On power bounded operators, J. Funct. Anal. 68 (1986), 313-328. Zbl0611.47005
  3. [3] Yu. Lyubich and V. Matsaev, Operators with separable spectrum, in: Amer. Math. Soc. Transl. (2) 47 (1965), 89-129. Zbl0158.14602
  4. [4] B. Nagy and J. Zemánek, A resolvent condition implying power boundedness, Studia Math. 134 (1999), 143-151. Zbl0934.47002
  5. [5] O. Nevanlinna, Convergence of Iterations for Linear Equations, Birkhäuser, 1993. 
  6. [6] R. K. Ritt, A condition that l i m n → ∞ n - 1 T n = 0 , Proc. Amer. Math. Soc. 4 (1953), 898-899. Zbl0052.12501
  7. [7] J. G. Stampfli, A local spectral theory for operators IV; Invariant subspaces, Indiana Univ. Math. J. 22 (1972), 159-167. Zbl0254.47007
  8. [8] E. Tadmor, The resolvent condition and uniform power boundedness, Linear Algebra Appl. 80 (1986), 250-252. 

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