CLASS OF ADMISSIBLE PERTURBATIONS OF SPECIAL EXPRESSIONS INVOLVING COMPLETELY MONOTONIC FUNCTIONS

Abstract

In this article, a class \(\mathbf{M}_2\) of admissible perturbations of the special expression \(\mathbf{M}_0 = \sum_{k=0}^r c_k t^{α_k} D_t^{ρ_k}\) in the weighted space \(\mathcal{L}_ω ^2([1, \infty))\) will be presented. It will be shown that the operator \(ω^{\frac{1}{2}} \mathbf{M}_2 ω^{-\frac{1}{2}}\), where ω belongs to the family of completely monotonic functions, is an admissible perturbation of \(\mathbf{M}_0\) in the non-weighted space \(\mathcal{L}^2([1, \infty))\), and eventually preserves the essential spectrum and nullity of \(\mathbf{M}_0\) in that space. Our discussion will be limited only to special expressions with \(α_1 < ρ_1\).