Weighted Moore-Penrose Inverses Of Partitioned Adjointable Operators On Hilbert C~*-modules

In this paper, we will introduce and study properties and general expressions of the weighted Moore-Penrose inverse AMN(?) for an adjointable operator A between Hilbert C*-modules. When A=(Aij) is 1×2 or 2×2 partitioned, the general expressions for AMN(?) in terms of the individual blocks Aij are studied. In the 2×2 partitioned case, an approach to constructing the Moore-Penrose inverse from the non-weighted case to the weighted case is provided by operator theory, and some results known for matrices are generalized in the general setting of Hilbert C*-module operators.The main parts of the paper are as follows:In Chaper 1, we introduce some basic knowledge about Hilbert C*-modules and adjointable operators. Some results known for matrices are generalized, such as propo-sitions 1.2.3,1.2.4 and 1.2.5.In Chaper 2, in the setting of Hilbert C*-module operators, we study expressions of the weighted Moore-Penrose inverse AMN(?) of an adjointable operator A. By using Sherman-Morrison-Woodbury formula, we obtain the relationship between AMN1(?) and AMN2(?), and thus obtain the general expression of AMN(?).In Chaper 3, we use the technique of constructing a commutative diagram to analyze 2×2 partitioned adjointable operators. In addition, the expression for the matrix are generalized to the case of the 2×2 partitioned adjointable operator. At last, the general expressions are given from the nonweighted case to the weighted case.In Chaper 4, we give derivations of the general expression for A(?),when A is a positive definite operator. Then we give detailed proofs of general expressions for Moore-Penrose inverses of a general operator.