Some characterizations of cone preserving Z-transformations

M. Seetharama Gowda (), Yoon J. Song () and K. C. Sivakumar ()
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M. Seetharama Gowda: University of Maryland, Baltimore County
Yoon J. Song: Soongsil University
K. C. Sivakumar: Indian Institute of Technology Madras

Annals of Operations Research, 2020, vol. 287, issue 2, No 9, 727-736

Abstract: Abstract Given a proper cone K in a finite dimensional real Hilbert space H, we present some results characterizing $$\mathbf{Z}$$Z-transformations that keep K invariant. We show for example, that when K is irreducible, nonnegative multiples of the identity transformation are the only such transformations. And when K is reducible, they become ‘nonnegative diagonal’ transformations. We apply these results to symmetric cones in Euclidean Jordan algebras, and, in particular, obtain conditions on the Lyapunov transformation $$L_A$$LA and the Stein transformation $$S_A$$SA that keep the semidefinite cone invariant.

Keywords: Proper cone; $$\mathbf{Z}$$ Z and Lyapunov-like transformations; Cone invariance; Irreducible cone; 47L07; 17C20; 90C33 (search for similar items in EconPapers)
Date: 2020
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DOI: 10.1007/s10479-017-2439-x

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