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.
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H. H. Bauschke, S. G Kruk, The method of reflection-projection for convex feasibility problems with an obtuse cone, J. Optimiz. Theory Appl., 120 (2004), 503–531.
A. Ben-Israel, T. N. E. Greville, Generalized Inverses: Theory and Applications, 2nd edn. Springer-Verlag, New York (2003).
A. Berman, R. J. Plemmons, Eight types of matrix monotonocity, Lin. Alg. Appl., 13, (1976), 115–123.
A. Berman, R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Classics in Applied Mathematics, SIAM, Philadelphia (1994).
A. Cegielski, Obtuse cones and Gram matrices with non-negative inverse, Lin. Alg. Appl., 335, (2001), 167–181.
L. Collatz, Aufgaben monotoner Art, Arch. Math., 3, (1952), 366–376.
L. Collatz, Functional Analysis and Numerical Mathematics, Academic, New York (1966).
M. I. Gil, On positive invertibility of matrices, Positivity., 2, (1998), 165–170.
M. I. Gil, Stability of Finite and Infinite dimensional systems, Kluwer Academic, Dordrecht (1998).
C. W. Groetsch, Generalized Inverses of Linear Operators: Representation and Approximation, Mercel Dekker, New York (1974).
W. J. Kammerer, R. J. Plemmons, Direct iterative methods for least-squares solutions to singular operator equations, J. Math. Anal. Appl., 49, (1975), 512–526.
S. H. Kulkarni, K. C. Sivakumar, Three types of operator monotonicity, J. Anal., 12, (2004), 153–163.
T. Kurmayya, K. C. Sivakumar, Nonnegative Moore–Penrose inverses of Gram operators, Lin. Alg. Appl., 422 (2007), 471–476.
T. Kurmayya, K. C. Sivakumar, Nonnegative Moore–Penrose inverses of Hilbert space operators, Positivity, DOI 10.1007/s11117-007-2173-8.
O. L. Mangasarian, Characterizations of real matrices of monotone kind, SIAM. Rev. 10, (1968), 439–441.
A. Novikoff, A characterization of operators mapping a cone into its dual, Proc. Amer. Math. Soc., 16, (1965), 356–359.
J. E. Peris, A new characterization of inverse-positive matrices, Lin. Alg. Appl., 154–156, (1991), 45–58.
K. C. Sivakumar, Nonnegative generalized inverses, Ind. J. Pure and Appl. Math., 28, (1997), 939–942.
K. C. Sivakumar, Range and group monotonicity of operators, Ind. J. Pure and Appl. Math., 32, (2001), 85–89.
K. C. Sivakumar, Applications of nonnegative operators to a class of optimization problems, Banach Center Publications (2008, to appear).
K. C. Sivakumar, J. Mercy Swarna, Some results on a class of optimization spaces, In: Gervasi, et al. (eds.) Lecture Notes in Computer Science, 3483, (2005), 1341–1348.
M. R. Weber, On the positiveness of the inverse operator, Math. Nachr., 163, (1993), 145–149. Erratum, Math. Nachr., 171, (1995), 325–326.
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Sivakumar, K.C. A new characterization of nonnegativity of Moore-Penrose inverses of Gram operators. Positivity 13, 277–286 (2009). https://doi.org/10.1007/s11117-008-2167-1
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DOI: https://doi.org/10.1007/s11117-008-2167-1