Hyponormal measurable operators affiliated with a semifinite von Neumann algebra. IV

Abstract

Let τ 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 τ-integrable operator B is found under which the operator A + B is normal. For an operator whose square is τ -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.