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

Let be a faithful normal semifinite trace on a von Neumann algebra M of operators. For a normal operator A in M, a condition on a -integrable operator B is found under which the operator A+B is normal. For an operator whose square is -integrable, equivalent conditions for its normality are established in terms of trace inequalities. For an operator in 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 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.