Let $ \tau $ be a faithful normal semifinite trace on a von Neumann algebra $ {\mathcal{M}} $ of operators.For a normal operator $ A $ in $ {\mathcal{M}} $ , a condition on a $ \tau $ -integrable operator $ B $ is found under which the operator $ A+B $ is normal.For an operator whose square is $ \tau $ -integrable, equivalent conditions for its normality are established in terms of trace inequalities.For an operator in $ {\mathcal{M}} $ , a criterion for hyponormality is found in terms of trace inequalities.It is shown that, given an arbitrary natural $ n $ , the power $ (PQ)^{n} $ of the product of projections $ P $ and $ Q $ in $ {\mathcal{M}} $ is hyponormal if and only if $ PQ=QP $ .Operator inequalities are obtained for powers of hyponormal contractions.It is shown that every natural power of a hyponormal partial isometry is a hyponormal partial isometry with the same initial space.
