Theses

Abstract:

The objective of this research work is to study the Moore-Penrose inverse of sums or differences of bounded linear operators on Hilbert spaces. Toward this, we consider the classical sum as well as a more generalized version called sub-direct sum. One of the main applications is to study the non-negativity of the Moore-Penrose inverse of these sums. We consider a related concept called the operator partial order which is defined on the algebra of all bounded linear operators on a Hilbert space.

Acknowledgements:   

This thesis work would not have been possible without my supervisor Dr. K. C. Sivakumar whos genuine care and encouragement throughout these years have helped me to complete this thesis. Further thanks are to Prof. S. H. Kulkarni, Prof. P. V. Subrahmanyam, Prof. Choudum, and Doctoral Committee members Prof. P. Veeramani (MA), Prof. M. V. Satyanarayana (PH) and Prof. Sanjay Kumar (CH) for their valuable suggestions and comments. Also I extend my sincere thanks to the thesis external examiner Prof. D. Bahuguna (IIT Kanpur, India).

Abstract:

The objective of this research work is to study Toeplitz matrices through circulants. Toeplitz matrices are matrices having constant entries along their diagonals. Problems modeled by Toeplitz matrices are: the numerical solution of certain differential equations, and certain integral equations; the computation of spline functions; time series analysis; signal and image processing; Markov chains and queuing theory; polynomial and power series computations. A circulant matrix is a special case of Toeplitz matrix where each row is a right cyclic shift of the row above it. The circulant matrix is fully specified by entries in the first row. The goal of this research work is to study a special case of Szego’s asymptotic eigenvalue distribution theorem which is the most famous and most important result describing the behavior of the eigen values as n goes to infinity..

Acknowledgements:   

This thesis work would not have been possible without my supervisor Dr. M.N.N. Namboodiri. Further thanks are to Prof. A. Krishnamoorthy, CUSAT, India.

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