Search

For each pair of bounded linear operators A and B on a Hilbert space H , let ‖ A − B ‖ be their operator distance. Given an idempotent Q on H , consider the operator distances from Q to all of the projections on H. It is proved that there always exists a projection whose operator distance away from Q takes the maximum value. An example is constructed to show that such a projection may fail to be unique. [ABSTRACT FROM AUTHOR]

For each pair of matrices A and B with the same order, let ‖ A − B ‖ F denote their Frobenius distance. This paper deals mainly with the Frobenius distances from projections to an idempotent matrix. For every idempotent Q ∈ C n × n , a projection m (Q) called the matched projection can be induced. It is proved that m (Q) is the unique projection whose Frobenius distance away from Q takes the minimum value among all the Frobenius distances from projections to Q , while I n − m (Q) is the unique projection whose Frobenius distance away from Q takes the maximum value. Furthermore, it is proved that for every number α between the minimum value and the maximum value, there exists a projection P whose Frobenius distance away from Q takes the value α. Based on the above characterization of the minimum distance, some Frobenius norm upper bounds and lower bounds of ‖ P − Q ‖ F are derived under the condition of P Q = Q on a projection P and an idempotent Q. [ABSTRACT FROM AUTHOR]

In this paper, we extend the operations of parallel addition A:B and parallel subtraction A ÷ B from the cone of bounded nonnegative self-adjoint operators to the linear bounded operators on a Hilbert space. Some conditions for the relations A † : B † = (A + B) † , B = (A : B) ÷ A , (A C) ÷ (B C) = (A ÷ B) C , A ÷ B = (P (A † − B †) P) † , B = A : (B ÷ A) , (C A) : (C B) = C (A : B) to be true are studied and the solution of the equation A:X = B is investigated. Moreover, some relationships between the parallel addition and subtraction of projections are obtained. [ABSTRACT FROM AUTHOR]

The basic properties of the parallel sum were developed for nonnegative matrices in finite-dimensional spaces. However, without suitable restrictions, very little of the preceding theories will hold for bounded linear operators A and B acting in Hilbert space. In this paper, generalization to non-selfadjoint operators is considered and various properties of parallel sum A : B = A (A + B) † B are given. Meanwhile, the common upper and lower bounds of positive operators by using the parallel sum are given and some relationships between the parallel sum of projection operators are obtained. [ABSTRACT FROM AUTHOR]