The Invariance Properties And Partial Order Of Operator Products Involving Generalized Inverses

In this thesis, we investigate the invariance properties of the bounded linear operator product ABCC^1,…"B^1,…A^1,…ABC and built the relationship between these invariance prop-erties and the mixed-type reverse order laws C{1,…}B[1,…}A{1,…}（?）（ABC）{1} for cor-responding generalized inverses on Hilbert spaces. Furthermore, we investigate the invari-ance properties of the bounded linear operator products AC^1,2,3D, AC^1,2,4B^1,2,4D, AC^1,3,4B^1,3,4)D and present the equivalent conditions for the invariance properties. Also, we study the range inclusion invariance properties of the operator product involving general-ized inverses and establish the relationship between invariance properties and its range under some equivalent conditions. Finally, the thesis study the partial order of operator involving generalized inverses and present some equivalent conditions relating Sharp order.