Stability Properties Characterizing the Spectra of Operators on Banach Spaces

✍ Scribed by S.Z. Huang

• Publisher: Elsevier Science
• Year: 1995
• Tongue: English
• Weight: 761 KB
• Volume: 132
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.1995.1109

✦ Synopsis

Let (A \in \mathscr{L}(E)) be a contraction. The famous Katznelson-Tzafriri theorem [11. Theorem 1] states that the spectral condition (\sigma(A) \cap \Gamma \subseteq{1}) is equivalent to the convergence of the orbit (\left{A^{n}(A-I): n=1,2, \ldots\right}) in norm to zero. Assume that the orbit (\left{A^{n}(A-I): n=1,2, \ldots\right}) is relatively compact in (\mathscr{L}(E)). Is there a spectral condition equivalent to this compactness? Such problems are studied for strongly continuous bounded representations of locally compact. abelian semigroups of linear operators on Banach spaces. A 1995 Academic Press. [nc.