Dissipative operators on Banach spaces

A bounded linear operator T on a Banach space is said to be dissipative if etT 1 for all t 0. We show that if T is a dissipative operator on a Banach space, then: (a) limt→∞ etT T =sup{|λ|: λ ∈ σ(T ) ∩ iR}. (b) If σ(T ) ∩ iR is contained in [−iπ/2, iπ/2], then lim t→∞ etT sin T = sup |sin λ|: λ ∈ σ(T ) ∩ iR . Some related problems are also discussed. © 2007 Elsevier Inc. All rights reserved.