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

Gil` M. I.

Journal of Applied Analysis

2013

Vol. 19, nr 2

231--246

We consider the operator T defined by (T f)(x)=(Sf)(x)+q(x)f(x), x ∈ Ω, where Ω ⊂ Rn is an unbounded domain, S is a positive definite selfadjoint operator defined on a domain Dom (S) ⊂ L2(Ω) and q(x) is a bounded complex measurable function with the property Im q(x) ∈ Lν(Ω) for a ν ∈ (1, ∞). We derive an estimate for the norm of the resolvent of T. In addition, we prove that T is invertible, and the inverse operator T-1 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 T by the multiplicative integral is established. As examples, we consider the Schrödinger operators on the positive half-line and orthant.