Study On The Completion Problems Of Operator Partial Matrices

Motivated by the commutant lifting theory, interpolation theory and control theory, recently, there is a growing interest in the study of the completion problems of operator partial matrices. Under this context, this thesis studies on few topics of the completion problems of operator partial matrices and gets some innovative results. Chapter 3 and chapter 4 are the main parts of this paper, chapter 3 mainly studies the weak completion problems of operator partial matrices on Hilbert spaces, First it discusses the weak right (left) invertible completion of operator partial ma-trices which is new, then it deepens the already results on Hilbert spaces such as the weak invertible completion of operator partial matrices and the weak Fredholm completion of operator partial matrices. Chapter 4 mainly studies the strong com-pletion problems of operator partial matrices on Banach spaces, extends the series results on Hilbert spaces to Banach spaces such as the strong invertible completion of operator partial matrices and the strong Fredholm completion of operator partial matrices, moreover, it discusses the strong Weyl completion problems of operator partial matrices on Banach spaces which is new and not having been discussed on Hilbert spaces.