Dissipative operators on Banach spaces

A bounded linear operator T   on a Banach space is said to be dissipative if ‖etT‖⩽1‖etT‖⩽1 for all t⩾0t⩾0. We show that if T is a dissipative operator on a Banach space, then:(a)limt→∞‖etTT‖=sup{|λ|:λ∈σ(T)∩iR}.(b)If σ(T)∩iRσ(T)∩iR is contained in [−iπ/2,iπ/2][−iπ/2,iπ/2], thenlimt→∞‖etTsinT‖=sup{|sinλ|:λ∈σ(T)∩iR}. Some related problems are also discussed.