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

Volume 134 / 1999

Yu Lyubich Studia Mathematica 134 (1999), 153-167 DOI: 10.4064/sm-134-2-153-167


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, $arccos(C^{-1}) < δ < π/2$. 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.


  • Yu Lyubich

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