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Theorems of Katznelson-Tzafriri type for contractions. (English) Zbl 0723.47013
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)], Y. Katznelson and L. Tzafriri [J. Funct. Anal. 68, 313-328 (1986; Zbl 0611.47005)], and G. R. Allan, A. G. O’Farrell and T. J. Ransford [Bull. London Math. Soc. 19, 537-545 (1987; Zbl 0652.46041)].

MSC:
47A60 Functional calculus for linear operators
47A45 Canonical models for contractions and nonselfadjoint linear operators
46J10 Banach algebras of continuous functions, function algebras
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