A new characterization of nonnegativity of Moore-Penrose inverses of Gram operators

Abstract

In this article, a new characterization for the nonnegativity of Moore-Penrose inverses of Gram operators over Hilbert spaces is presented. The main result generalizes the existing result for invertible Gram operators. Mathematics Subject Classification (2000). 15A48, 15A09.

Moore-Penrose inverses of Gram matrices leaving a cone invariant in an indefinite inner product space

“…In this article, we avoid the notion of pairwise acuteness of cones and characterize the Moore-Penrose inverses of Gram matrices leaving a cone invariant in the approach of Sivakumar [17]. These results generalize the existing results of Sivakumar [17] in the finite dimensional setting from Euclidean spaces to indefinite inner product spaces. In this section, we prove a series of results that lead up to the main theorem (Theorem 3.8).…”

“…In particular, nonnegativity of the inverse of Gram operators has been studied in connection with certain optimization problems [4], where a characterization is proved. This characterization has been extended to operators between Hilbert spaces [7] and [17]. In the latter, a completely new approach was proposed.…”

“…In the latter, a completely new approach was proposed. The sole aim of the present work is to extend this characterization of nonnegativity of the Moore-Penrose inverse of a Gram operator in an indefinite inner product space with the indefinite product of matrices, adopting the approach taken as in [17]. Here, nonnegativity should be interpreted in terms of taking one cone into another.…”

Infinite Matrices and Their Recent Applications

“…The last section considers a class of operators that are more general than M-operators. In particular, we review results relating to the nonnegativity of the Moore-Penrose inverse of Gram operators over Hilbert spaces, reporting the work of Kurmayya and Sivakumar [61] and Sivakumar [125]. These results find a place here is due to the reason that they extend the applicability of results for certain subclasses of M-matrices to infinite dimensional spaces.…”

Nonnegative Generalized Inverses and Interval Linear Programs

“…An interesting source of reference on this subject is the monograph by Berman and Plemmons [4]. Various aspects of nonnegative generalized inverses in infinite dimensions can be found in [5,11,12,17] and the references cited therein. Our objective here is to highlight the interplay between nonnegativity of various generalized inverses and interval linear programs.…”