An estimate for the resolvent of a non-self-adjoint differential operator on the half-line

We consider the operator defined by (T₀y)(x) = -y" + q(x)y (x > 0) on the domain Dom (T0) = { f ∈ L²(0,∞) : f" ∈ L²(0,∞), f (0) = 0}. Here q(x) = p(x) + ib(x), where p(x) and b(x) are real functions satisfying the following conditions: b(x) is bounded on [0,8), there exists the limit b0 := lim_x→∞ b(x) and b(x) - b₀ ∈ L²(0,8). In addition, inf_x p(x) > sup_x |b1(x)|. We derive an estimate for the norm of the resolvent of T0, as well as prove that (T₀ - ib₀I)^-1 is a sum of a normal operator and a quasinilpotent one, and these operators have the same invariant subspaces.