EP And Hypo-EP Properties Of Operator Matrices On Hilbert C*-modules

This dissertation mainly deals with the positive semi-definiteness,(weighted)MoorePenrose inverse,(weighted)EP and(weighted)hypo-EP property of operator matrices on Hilbert C*-module setting,based on the space decomposition method,generalized inverse and generalized Schur complement.More precisely,the positive semi-definiteness,(weighted)Moore-Penrose inverse,(weighted)EP and(weighted)hypo-EP properties of operator matrices are described by the corresponding properties of their operator entries.The main results of this dissertation are as follows:First,using the space decomposition method,we study the expression of the MoorePenrose inverse of the upper triangular block operator matrix(?),and discuss the conditions under which H0 admits such Moore-Penrose inverse expression.Based on these results,the necessary and sufficient conditions for H0 to be EP and hypo-EP are discussed.As applications,the corresponding properties of upper triangular Hamiltonian modular operator(?)are given.Next,the EP and hypo-EP properties of the operator matrix(?)are discussed.Some necessary and sufficient conditions for H to be EP and hypo-EP are given,based on the generalized inverse and generalized Schur complement,and some special circumstances are also analyzed.Moreover,the applications of EP and hypo-EP operators to operator equations are studied.Then,the positive semi-definiteness of self-adjoint operator matr(?)are considered.Some properties of positive semi-definite operator matrix H are studied,and the necessary and sufficient conditions for the self-adjoint operator matrix H to be positive semi-definite are obtained by using the generalized Schur complement which is related to inner inverse.Finally,the weighted Moore-Penrose inverse,weighted EP and weighted hypo-EP properties of operator matnx(?)are studied.The definition of weighted hypo-EP operator is proposed,and some equivalence conditions and related properties of weighted hypo-EP operators are described.Using weighted Moore-Penrose inverse and weighted generalized Schur complement,the necessary and sufficient conditions for the weighted Moore-Penrose inverse of H to have Banachiewicz-Schur form are studied.The necessary and sufficient conditions for H to be weighted EP and weighted hypo-EP are further established.Besides,the application of weighted EP operator in operator equations is given.