## Date of Submission

3-28-1966

## Date of Award

3-28-1967

## Institute Name (Publisher)

Indian Statistical Institute

## Document Type

Doctoral Thesis

## Degree Name

Doctor of Philosophy

## Subject Name

Computer Science

## Department

Research and Training School (RTS)

## Supervisor

Rao, C. Radhakrishna (RTS-Kolkata; ISI)

## Abstract (Summary of the Work)

The theory of positive matrices (matrioco with non- negative entries) nnd the the ory of positive operetors are used extensively in the study of vibrationa of mechanical eysteme l16], stochaatic proceuses (36] nd mathematical economics ((9] and L20].) It is well known from the claseical theoreme of Perron and Frobenius ((13), [14) and (25] that any non-eingul ar squere matrix with non-negative entries has a positive cigenvalue which ts maximal in modulus among al1 the eigenvalues of the matrix. Further for this positive eigenvalue, it has a non-null eigenvector with all components non-negative. If the matrix is also irroduc lble11) it wae shown by Probenius that the positive et genvaluc, maximal in modulus, 1s algebraically and geometrically simple. Further if there are exactly h eigenvalues wi th an abeolute value equal to the above positive eigenvalue, any r, then they are precisely the roots of the equation ah-rh= 0. The theorem of Perron-Frobehius wan cstablishcd by Jentsch (17) for integral equationeÉ¸(s) = ÊŽ Êƒ abk(t, s) É¸ (É›) ds with positive kernel (1,e k(t, s) > o Sur a â‰¤ t, a â‰¤ b).When the positive kernel k(t, a) 1s continuous it was shown by Jentach, that there cxiste a positive continuous solution É¸ for the above integral eguation for a positive parameter ÊŽ. Purther the parame ter ÊŽ ie nimple and greatest in modulus of the root s of the Frodholmb determinant.Since the ponitive orthant in a cone 12) and since matriccs with non-negative entries leave this cone invariant, Krein and Rutman (23) extended the theorema of Perron and Frobenius to arbitrary real Benanh apacee, by studying the apectral theory of operatora that leave a cone 1nvariant. Since they are the infinite analogues of matrices with positive ontries, they are aleo knowm na penitive operators.When a conc in a real Banach space han a dense linear hull and when a compact linear operator is pocitive with respect to this cone with positive apectral radius, it was shown by Krein and Rutman (23], that the spcctral radlus i9 actually an eigenvalue of the operator and it has a non- null eigenvee tor in the cono. Further in the case of cones with interior if the compact lingar operator is strongly positive, they proved thet the spectral radius is a simple eigenvalue end the eigenvector in the cone is a simple eigenvector. It was also proved by them that if a positivo line ar operator, positive with respect to a normal (41, has a poaitive eigenvalus with an ei.genvector in the cone intorior of the cono, then this cigenvalue io actually the apectral radius of the operator. Suppose if we go back to their motivation, then we find that the positive orthant, instead of hetng viewod as a cone, cean an woll bo viuwed as a closed eonvox sat with origin as an extrome point and square matricus with non- negative cntrics lCAve this convex get invariant. Thue the following question naturally arisas.

## Control Number

ISILib-TH77

## Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

## DOI

http://dspace.isical.ac.in:8080/jspui/handle/10263/2146

## Recommended Citation

Raghavan, T. E.S. Dr., "Extensions of the Theory of Positive Operators and Their Relationship to Minimax Games." (1967). *Doctoral Theses*. 124.

https://digitalcommons.isical.ac.in/doctoral-theses/124

## Comments

ProQuest Collection ID: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqm&rft_dat=xri:pqdiss:28842900