The Banach algebra generated by a contraction

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Let T T be a contraction on a Banach space and A T A_{T} the Banach algebra generated by T T . Let σ u ( T ) \sigma _{u}(T) be the unitary spectrum (i.e., the intersection of σ ( T ) \sigma (T) with the unit circle) of T T . We prove the following theorem of Katznelson-Tzafriri type: If σ u ( T ) \sigma _{u}(T) is at most countable, then the Gelfand transform of R ∈ A T R\in A_{T} vanishes on σ u ( T ) \sigma _{u}(T) if and only if lim n → ∞ ‖ T n R ‖ = 0. \lim _{n\rightarrow \infty }\left \Vert T^{n}R\right \Vert =0.