A Study on Certain Positivity Classes of Operators in Hilbert Spaces

Abstract

In mathematical optimization theory, the linear complementarity problem, which is stated as, given a vector q in a finite-dimensional real vector space and an n×n real matrix A, then, finding a vector x that satisfies the system of inequalities x ≥ 0, q+Ax ≥ 0, xT(q+Ax) = 0, plays a vital role in many areas such as bimatrix game theory, mar ket equilibrium, computational complexity, and many more. The nature of the solution of the linear complementarity problem can be discussed with the help of the matrix A involved in the problem. The sign-reversing is a property of matrices along with a given vector, which is stated as an n×n matrix A reverses the sign of a vector x in an n-dimensional real vector space if it satisfies xi(Ax)i ≤ 0 for all index i. The con cept of sign-reversing is a useful tool to identify and characterize certain matrix classes involved in linear complementarity problems. The sign-reversing set of a matrix A is defined as {x : xi(Ax)i ≤ 0, ∀i}. In this thesis, we characterize the sign-reversing set of an arbitrary square matrix A in terms of the null spaces of the matrices DA−A−D, where D is a diagonal matrix such that 0 ≤ D≤I. The matrices which have convex sign-reversing sets include large classes of matrices; we discuss a subclass of matrices in which the convexity of the sign-reversing set is characterized. Areal square matrix A is called a P-matrix if all its principal minors are positive. In 2016, Kannan and Sivakumar extended the notion of P-matrix to infinite-dimensional Banach spaces relative to a given Schauder basis by using the sign non-reversal prop erty of matrices. Motivated by their work, we discuss P-operators on separable real Hilbert spaces with the help of the inner product structure of the Hilbert spaces. We also investigate P-operators relative to various orthonormal bases. In this thesis, we define the concept of a sign-reversing set for operators on separa ble Hilbert spaces with the help of the inner product structure of the Hilbert spaces rel ative to a given orthonormal basis. We also generalize some special classes of matrices to operators in infinite-dimensional Hilbert spaces with the help of the sign-reversing property of operators. The thesis also studies the spectral theory of certain positive operators under consideration.