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2 Journal of Applied Analysis

2 Gil` M. I.

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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.

A priori solution estimates for nonlinear vector integral equations

Gil` M. I.

Journal of Applied Analysis

2003

Vol. 9, nr 2

187-200

Nonlinear vector integral equations are considered. Solution estimates and solvability conditions are derived. Applications to the periodic boundary value problem are also discussed. Under some restrictions our results improve the well-known ones. The main tool in the paper is the recent estimates for the resolvent of Hilbert-Schmidt operators.