Date of Submission


Date of Award


Institute Name (Publisher)

Indian Statistical Institute

Document Type

Doctoral Thesis

Degree Name

Doctor of Philosophy

Subject Name



Theoretical Statistics and Mathematics Unit (TSMU-Delhi)


Parthasarthy, K. R. (TSMU-Delhi; ISI)

Abstract (Summary of the Work)

The study of spectra occupies a central place in the theory of linear operators. One part of this study consists of obtaining a detailed and complete knowledge of the spectrum of a given lincar operator. The other part - perturbation theory - consists of using this knowledge to obtain information about the spectra of nearby operators.Apart from the intrinsic mathematical interest it has, perturbation theory is of great importance in the study of several physical problems. In fact, the theory came into existence with the work of Rayleigh on sound waves and that of Schrodinger on quantum mechanics. Later, their results were put on a firm mathematical ground by Rellich and developed further by him and several other mathematicians. A detailed account of these results may be found in Kato [12].When the underlying linear space is finite-dimensional the spectrum of an operator consists of its eighevalues. Though the finite - dimensional theory is simpler it is not trivial. (See the books by Kato (12), Rellich [17) and Baumgartel [1)). The main problems of the subject can be classifie into three types of questions which have bearings upon each other. These are If z + A(z) is a holomorphic operator valued map based on a 1. complex domain, are the eigenvalues, eigenprojections and eigernilpotents holomorphic function of z ? This question has been discussed in detail in the three books mentioned above.2. What arc the power series expansions for the objects in Question 1 ? What are the radii of convergence of these series and what are the error - estimates when only a finite number of terms of these series are taken into account ? This question has also been considered in detail in the books refered to above.3. If the operators A and B are close to each other how close are the eigenvalues and eigenvectors of A to those of B ?This thesis is concerned primarily with the third question about eigenvalues. We introduce some new methods for the study of this problem and using them obtain explicit estimates for 'the distance between the cigonvalues of two operators,Apart from the viewpoint of perturbation theory this problem is of interest and importance in computational linear algebra and in approximation theory. Heme the problems can be statod in the following terms. If the entries of a matrix are known approximately, to what degree of approximation are the eigenvalues known ? If the entries of a matrix can be ascertained only upto a certain decimal place, to how many places should íts eigenvalues be computed ? If a sequence A of operators converges to A, how fast do the eigenvalues of A. converge to those of A ? These questions have boen analysed by several authors. We give below a brief summary of the prominent results gormane to our study. These are given not in the chronological order in which they were obtained but in the ascending order of generality to which they pertain.


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