Positivity 13 (2009), 277–286
2008 Birkh¨auser Verlag Basel/Switzerland
1385-1292/010277-10, published online April 30, 2008
A new characterization of nonnegativity
of Moore-Penrose inverses
of Gram operators
K. C. Sivakumar
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.
Keywords. Gram operator, nonnegative Moore–Penrose inverse, acute cones,
A square real matrix A is called monotone if Ax ≥ 0 ⇒ x ≥ 0. Here x =(x
) ≥ 0
means that x
≥ 0 for all i. Collatz  has shown that a matrix is monotone if and
only if it is invertible and the inverse is nonnegative. We refer the reader to for
details regarding certain properties of monotone matrices and their applications
to iterative solutions of systems of linear equations arising out of applying ﬁnite
diﬀerence approximations to a certain second order boundary value problem.
The notion of monotonicity has undergone generalizations along several direc-
tions. We mention only some of the more recent works in the literature, beginning
with the case of classical inverse. Gil gave suﬃcient conditions on the entries of a
matrix A in order for A
to be nonnegative . Peris  adopted a novel approach
where he characterized nonnegativity of the inverse in terms of splittings of a cer-
tain type. Extensions of the notion of monotonicity to inﬁnite dimensional spaces,
for invertible operators were studied by Gil  who gave suﬃcient conditions under
which the inverse of an inﬁnite matrix is entry wise nonnegative. For operators
over normed vector lattices, extensions of some of the results in , were reported
by Weber , with an ingenious proof of the main theorem (Theorem 1, ).
An extension of the notion of monotonicity to characterize nonnegativity of
generalized inverses in the ﬁnite dimensional case seems to have been ﬁrst accom-
plished by Mangasarian . Berman and Plemmons made extensive contributions