Pairs Of Positive Operators Research Articles

Cancellability and regularity of operator connections with applications to nonlinear operator equations involving means

An operator connection is a binary operation assigned to each pair of positive operators satisfying monotonicity, continuity from above, and the transformer inequality. A normalized operator connection is called an operator mean. In this paper, we introduce and characterize the concepts of cancellability and regularity of operator connections with respect to operator monotone functions, Borel measures, and certain nonlinear operator equations. As applications, we investigate the existence and the uniqueness of solutions for operator equations involving various kind of operator means.

Positivity, Betweenness, and Strictness of Operator Means

An operator mean is a binary operation assigned to each pair of positive operators satisfying monotonicity, continuity from above, the transformer inequality, and the fixed-point property. It is well known that there are one-to-one correspondences between operator means, operator monotone functions, and Borel measures. In this paper, we provide various characterizations for the concepts of positivity, betweenness, and strictness of operator means in terms of operator inequalities, operator monotone functions, Borel measures, and certain operator equations.

Singular values of compressions, restrictions and dilations

For an operator A and a subspace E, denote by A:E the restriction of A to E and by AE the compression of A to E. A pair (S,T) of hermitian operators is said to be monotone if there exist two nondecreasing functions f, g and a hermitian Z such that S=f(Z) and T=g(Z). Using this notion: (1) We give a simple analytic characterisation of invertible operators X such that Sing(A:X(E))⩾Sing(A:E) for all subspaces E, where Sing(·) stands for the sequence of singular values. Equivalently, if A is hermitian we characterise invertible operators X satisfying Eig(AX(E))⩾Eig(AE) for all subspaces E, Eig(·) standing for the sequence of eigenvalues. (2) We prove many inequalities involving monotone pairs of positive operators. For instance we show that detAE·detBE⩽det(AB)E. In some other circumstances, the opposite inequality holds. We also show that pairs of hermitians can be dilated into monotone pairs.

Characterization of parallel subtraction

Parallel subtraction is an operation defined on pairs of positive operators. In terms of electrical networks, one may pose the following problem: Given an electrical network, represented by a specified positive operator, determine the set of positive operators which when connected in parallel with the specified operator yield another prescribed operator. The set of solutions of this electrical network problem is shown to have a minimum. The minimum is termed "the parallel difference of the fixed operators," and the operation is termed "parallel subtraction." The parallel difference is used to obtain explicit error estimates for an iteration procedure which approximates the geometric mean of positive operators. This concept of the geometric mean reduces to the square root of the product of the operators if the operators commute. Finally, by using the geometric mean, an operator version of the Gaussian mean is presented.

Stability of systems with sector nonlinearities: A comparison of various inequalities

A common method of deriving stability conditions of the type involves two steps. 1) the feedback equation is transformed into a new equation containing a pair of positive operators 2) an appropriate multiplier is introduced. Depending on which of these steps is performed first, two different results are obtained, and it has not been clear in the past whether either result is preferable to the other. It is shown here that, under suitable hypotheses, it is preferable to perform step 1) first, i.e., a better stability condition is obtained if the transformation into positive operators is performed before the multiplier is introduced.