Assume that $ A $ generates a contraction semigroup on a Hilbert space $ X $, and B is a bounded linear operator on $ X $. I want to show that $ A + B - 2|| B ||I $ with the domain equal to the domain of $ A $, generates a contraction semigroup on $ X $.
The Lumer-Philips theorem says that a linear operator $ A $ generates a contraction semigroup on $ X $ iff it is dissipative and $ \text{range}(I - A) = X $. It is not too hard to show that if $ A $ is dissipative, then so is $ A + B - 2 || B || I $. For this, I one can use the characterization $ A $ is dissipative iff $ || (aI- A) x || \geq a||x|| $ for $ x \in D(A), a > 0 $.
However, I have no clue how to show that $ \text{range}((1 + 2|| B ||)I - A - B) = X $ assuming $ A $ generates a contraction semigroup (so is a linear operator on $ X $) and $ B $ is a bounded linear operator on $ X $.