An estimate for the resolvent of a non-selfadjoint differential operator on an unbounded domain

We consider the operator defined by , , where is an unbounded domain, is a positive definite selfadjoint operator defined on a domain and is a bounded complex measurable function with the property for a . We derive an estimate for the norm of the resolvent of . In addition, we prove that is invertible, and the inverse operator is a sum of a normal operator and a quasinilpotent one, having the same invariant subspaces. By the derived estimate, spectrum perturbations are investigated. Moreover, a representation for the resolvent of by the multiplicative integral is established. As examples, we consider the Schrödinger operators on the positive half-line and orthant.