Abstract

For an arbitrary operator T on Hilbert space, we study the maps Φ : f(T) → f(T) and Φ : f(T) → f(T), where T and T are 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 T and T, and also the inclusion W(f(T))⁻∪W(f(T))⁻⊂ W(f(T))⁻ relating the numerical ranges of f(T), f(T), and f(T).

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