Two counterexamples in semigroup theory on Hilbert space

Abstract

There exist ( C 0 ) ({C_0}) semigroups T 1 ( t ) , T 2 ( t ) {T_1}(t),{T_2}(t) on Hilbert space with the following properties: T 1 {T_1} has a bounded generator and is uniformly bounded, but is not similar to a contraction semigroup. T 2 {T_2} is uniformly bounded, and there exists no scalar α \alpha such that e − α t T 2 ( t ) {e^{ - \alpha t}}{T_2}(t) is similar to a contraction semigroup.