Norm inequalities for sums and differences of positive operators Original Research Article

It is shown that if A and B are positive operators on a separable complex Hilbert space, and if short parallel·short parallel is any unitarily invariant norm, thenimage2short parallelAcircled plusBcircled plus0circled plus0short parallelless-than-or-equals, slantshort parallel(A−B)circled plus(A−B)circled plus0circled plus0short parallel+short parallelAcircled plusAcircled plusBcircled plusBshort parallel+short parallelA1/2B1/2circled plusA1/2B1/2circled plusA1/2B1/2circled plusA1/2B1/2short parallel.When specialized to the usual operator norm short parallel·short parallel, this inequality reduces toimagemax(short parallelAshort parallel,short parallelBshort parallel)−short parallelA1/2B1/2short parallelless-than-or-equals, slantshort parallelA−Bshort parallel.Related inequalities for sums and differences of positive operators are obtained, and applications of these inequalities to norms of self-commutators are also considered.