Numerical The Supremum Nature In The Child

The infimum and supremum of operators, and generalized inverse of operators are heated topics in operator theory and also have important value in both theory and application.The research of these subjects has related to pure and applied mathematics such as functional analysis, algebra, geometry, operator perturbation theory, matrix analysis, approximation theory and quantum physics etc. The research of this thesis focuses on the infimum and supremun of self-adjoint operators in a complex Hilbert space with respect to the Gudder order, the infimum and supremun of operators in a complex Hilbert space with respect to the *-order and theΓ-inverse of operators in a complex Hilbert space. By using the technique of operator blocks, the internal relations and constructions among operators can be found.The present thesis is divided into three chapters and organized as follows:In Chapter 1, using space decomposition and block operator matrices, we study the infimum and supremun of self-adjoint operators in a complex Hilbert space with respect to the Gudder order. For self-adjoint operators A and B, we show that there exists A∧_G B and establish the explicit representation of A∧_G B, where A∧_G B denotes the infimum of A and B associated with the Gudder order. We present necessary and sufficient conditions for which there exists A∨_G B and in this case we establish the explicit representation of A∧_G B, where A∨_G B denotes the supremum of A and B associated with the Gudder order.In Chapter 2, using space decomposition and block operator matrices, we study the infimum and supremun of operators in a complex Hilbert space with respect to the *-order. For operators A and B, we show that there exists A(?)B and establish the explicitrepresentation of A (?)B, where A(?)B denotes the infimum of A and B associated withthe *-order. We present necessary and sufficient conditions for which there exists A(?)Band in this case we establish the explicit representation of A(?)B, where A(?)B denotes the supremum of A and B associated with the *-order.In Chapter 3, using space decomposition and block operator matrices, we study theΓ-inverse of operators in a complex Hilbert space. For operators A, P and Q, we present necessary and sufficient conditions for which there exists A_P,Q⁺ and in this case we establish the explicit representation of A_P,Q⁺, where A_P,Q⁺ denotes theΓ-inverse of A associated with P and Q.