Geometry Structures Of Shorted Operators And Its Applications

The study of operator theory began in 20th century. Since it is usedwidely in mathematics and other subjects, it got rapid development at the beginningof the 20th century, Shorted operators and Schur complements have become hottopics in operator theory. Let A∈B(H)⁺ and S(?)H be a closed subspace of H,the shorted operator∑(S, A) of A to S has been defined as follows∑(S, A)=max{X∈B(H)⁺:X≤A and R(X)(?)S},where the maximum is taken for the natural order relation in B(H)⁺. The research ofthis thesis focuses on the geometry structures of Shorted operators, infimum of twopositive operators and the positive solutions to operator equations. The research onShorted operators brings out the following results: the geometry structure and someproperties of∑(S, A), the characterization of the compatible of (S, A). The infimumproblem of Hilbert space effects is when the infimum A∧B exists for A, B∈ε(H), inthis article we generalized the infimum problem to positive operators and containedthe sufficient and necessary condition when A∧B exists, here A, B∈B(H)⁺. Onthe research of operator equations, we obtain the necessary and sufficient conditionsin which the operator equation AX=C has positive solutions. Moreover, therepresentation of the general positive solutions to this equation is given.This paper contains four chapters.In chaper 1, we mainly introduce some notations, definitions and theorems, suchas the definitions of positive operator, spectrum, Shorted operator, compatibility,infimum of positive operators etc. Subsequently, we give some well-known theoremssuch as the rang inclusion theorem and spectral mapping theorem.In chapter 2, we use the block-operator matrix technique to study the shortedoperators. The relations of geometry structures between a positive operator and itsshorted operator are exposed. Besides, we investigate the compatibility of the pair(A, S), where A is a bounded linear self-adjoint operator on a Hilbert space H andS is a closed subspace of H. Particularly, when A is positive, we give a detailed characterization of the set P(A, S)={Q∈P:R(Q)=S^⊥, AQ=Q^*A}, where Pand S^⊥denote the set of all idempotents on H and the orthogonal complement ofS, respectively.In chapter 3, we investigate the infimum problem of two positive operatorsA, B∈B(H)⁺ by Shorted operators on an infinite dimensional Hilbert space, andobtain the sufficient and necessary condition in which A∧B exists. The main resultsas follows:(1). Let A∈B(H)⁺ and P be the orthogonal projection onto S. Then P∧Aexists if and only ifσ(∑(S, A))(?){0}∪[1,‖A‖] orσ(∑(S, A))(?)[0, 1].(2). Let A, B∈B(H)⁺, then the infimum of A, B A∧B exists if and only if∑(S₀, A) and∑(S₀, B) are comparable, where S₀=(?)∩(?).In chapter 4, We firstly characterize the Moore-Penrose generalized inverse ofthe operator matrix(?)in the condition R(A₁^*)∩R(A₂^*)={0}. Secondly,we discuss the conditions in which the operator equation AX=C has self-adjointsolutions and positive solutions, then obtain the sufficient and necessary conditionswhen the operator equation AX=C has self-adjoint solutions and positive solu-tions, respectively. Moreover, we get the equivalent condition in which the operatorequations AX=C and XB=D have common positive solutions.