Theorems of Katznelson-Tzafriri type for contractions
Let T be a Hilbert space contraction and fa function holomorphic on the unit disk and continuous on its boundary \(\Gamma\). The authors prove that \(\| f(T)T^ n\| \to 0\) (n\(\to \infty)\) if and only if \(f=0\) on \(\sigma\) (T)\(\cap \Gamma\). (The main theorem 2.10, proved for Banach space operators is, in fact, more general.) This result is related to some recent works by the first author [Lect. Notes Math. 975, 66-162 (1983; Zbl 0536.46041)], \textit{Y. Katznelson} and \textit{L. Tzafriri} [J. Funct. Anal. 68, 313-328 (1986; Zbl 0611.47005)], and \textit{G. R. Allan}, \textit{A. G. O'Farrell} and \textit{T. J. Ransford} [Bull. London Math. Soc. 19, 537-545 (1987; Zbl 0652.46041)].
scientific article; zbMATH DE number 4192454
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