Sum Of Positive Operators Research Articles

On form sums of positive operators

The purpose of the present note is to provide domain, kernel and range characterizations for the form sum of two positive selfadjoint operators. In addition, we establish a criterion for the closedness of the range of the form sum and give the Moore–Penrose pseudoinverse in this case.

Minimum-Error Discrimination Between Self-Symmetric Mixed Quantum State Signals

We derive some properties of minimum-error measurements for mixed quantum state signals where a density operator itself has a certain symmetry. In the first case, we show that if there exists a regular normal operator that commutes with each density operator of mixed state signals, then there exists an optimal measurement that consists of detection operators expressed as the direct sum of positive operators with supports in the eigenspaces of the regular normal operator. In the second case, we show that for abelian geometrically uniform state signals, if a matrix which represents one of the signals in a certain basis has all real elements, then the matrix which represents the corresponding detection operator of an optimal measurement in that basis also has all real elements. In addition, we discuss some examples of applications of these properties and derive analytical solutions of optimal measurements for special cases.

Norm Inequalities for Certain Operator Sums

We extend to larger classes of norms some inequalities concerning sums of positive operators and sums of operators having orthogonal ranges. These inequalities are useful in the theories of best approximation inC*-algebras and generalized inverses.