‎let $lambda_1,dots,lambda_n$ be positive real numbers such that‎‎$sum_{k=1}^n lambda_k=1$‎. ‎we prove that for‎‎any positive operators $a_1,a_2,cdots‎, ‎a_n$ in semifinite von‎‎neumann algebra $m$ with faithful normal trace that $t(1)

This work studies the spectral properties of certain unbounded selfadjoint operators by considering positive perturbations of such operators and the unitary equivalence of the perturbed and unperturbed transformations. Conditions are obtained on the unitary operators implementing this equivalence which guarantee that the selfadjoint operators have an absolutely continuous part.