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

Abstract

We consider the operator defined by \documentclass[12pt]{minimal}\begin{document}$(T_0y)(x)=-y^{\prime \prime }+ q(x)y\;\;(x>0)$\end{document}(T0y)(x)=−y′′+q(x)y(x>0) on the domain Dom (T0) = {f ∈ L2(0, ∈fty): f′′ ∈ L2(0, ∈fty), 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, ∞), there exists the limit b0 ≔ limx → ∞b(x) and b(x) − b0 ∈ L2(0, ∈fty). In addition, \documentclass[12pt]{minimal}\begin{document}$\inf _x p(x)> \sup _x |b_1(x)|$\end{document}infxp(x)>supx|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.