Complete contractivity of maps associated with the Aluthge and Duggal transforms

For an arbitrary operator T on Hilbert space, we study the maps Φ̃: f(T) → f(T̃) and Φ̂: f(T) → f(T̂), where T̃ and T̂are the Aluthge and Duggal transforms of T, respectively, and f belongs to the algebra Hol(σ(T)). We show that both maps are (contractive and) completely contractive algebra homomorphisms. As applications we obtain that every spectral set for T is also a spectral set for T̂ and T̃, and also the inclusion W(f(T̃))^- ∪ W(f(T̂))^- ⊂ W(f(T))^- relating the numerical ranges of f(T), f(T̃), and f(T̂).