Nonnegative Moore–Penrose inverses of Gram operators

Abstract

In this paper we derive necessary and sufficient conditions for the nonnegativity of Moore-Penrose inverses of unbounded Gram operators between real Hilbert spaces. These conditions include statements on acuteness of certain closed convex cones. The main result generalizes the existing result for bounded operators [11, Theorem 3.6].

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

“…Recently, a set of necessary and sufficient conditions for the nonnegativity of (A * A) † were given in Theorem 3.6, [13]. In this section, we prove a new characterization of nonnegativity of Moore-Penrose inverses of Gram operators between Hilbert spaces.…”

“…The main result generalizes the existing result of Novikoff for invertible Gram operators. We mention that the present work is independent of [13]. It will be interesting to study the possibility of obtaining a characterization that unifies all the known ones.…”

“…It is pertinent to mention that our main result here is motivated by this particular work of Novikoff. Finally, we mention that very recently, the characterization of nonnegativity of inverses of Gram matrices, was extended to Gram operators over infinite dimensional real Hilbert spaces (Theorem 3.6, [13]). …”

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

“…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.…”

Nonnegative Moore-Penrose inverses of operators over Hilbert spaces

“…Berman and Plemmons [2] generalized the concept of monotonicity in several ways. Recently, Kurmayya and Sivakumar [8] characterized nonnegativity of Moore-Penrose inverses of Gram operators in terms of obtuseness (or acuteness) of certain cones. Berman and Plemmons [2] and several others (See [9] and [10,11] and the references cited therein) have characterized monotonicity of matrices and operators using splittings.…”