Hyponormal Measurable Operators Affiliated with a Semifinite von Neumann Algebra. IV

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.