Approximation Of Aperators

The main purpose of this paper is to study Postive Approximants of Operators. Given an operator A, determine δ(A) = inf{||A- P||: P ≥ 0}and find a P for which the infimum is attained. We mainly resolve a prob-lem in Halmos[2]: for A = , we can compute P₀ =is a positive approximant of A, and from theory analysis, P₀ should be the unique approximant of A. But he hasn't proved this. We prove that P₀ is the unique positive approximant of A. For some special sorts A, for example, A is a normal operator, or an Hermitian operator ect. we study the unique condition of positive approximants. For other sorts of approximants of A, we study some. For example, for a normal operator A, its positive contracion approximants and its unique condition.