**Date & Time:** Wednesday, March 02, 2016, 16:00-17:00.

**Venue:** Ramanujan Hall

**Speaker:**T.E.S. Raghavan,

University of Illinois at Chicago, USA

**Title:** Perron-Frobenius Theorem and a theorem of
M.G. Krein on Positive Operators via Game Theory

**Abstract:** The minimax theorem of von Neumann can be applied to the matrix A − λI
where A is an irreducible matrix with nonnegative entries. The associated value function v(λ)
and the optimal mixed strategies for the two players for v(μ) = 0 for some μ turn out to be
the Perron root and the associated eigenvector. A theorem of Kaplanski shows that the game
is completely mixed and thus the algebraic simplicity of μ is a direct consequence of this game
theoretic result. For normal cones in real reflexive Banach spaces, the theorem of Krein on
positive operators do yield positive eigenvalue and an eigenvector in the cone. The proof uses
the general minimax theorem.