Asymptotic behavior of polynomially bounded operators

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Let T be a polynomially bounded operator on a complex Banach space and let A T be the smallest uniformly closed (Banach) algebra that contains T and the identity operator. It is shown that for every S ∈ A T , lim n → ∞ ‖ T n S ‖ = sup ξ ∈ σ u ( T ) | S ˆ ( ξ ) | , where S ˆ is the Gelfand transform of S and σ u ( T ) : = σ ( T ) ∩ Γ is the unitary spectrum of T; Γ = { z ∈ C : | z | = 1 } .