Please use this identifier to cite or link to this item: http://hdl.handle.net/10397/16662
Title: Linear operators and positive semidefiniteness of symmetric tensor spaces
Authors: Luo, ZY
Qi, LQ 
Ye, YY 
Keywords: Linear operator
SOS cone
Symmetric positive semidefinite tensor cone
Symmetric tensor
Issue Date: 2014
Publisher: Science in China Press
Source: Science China. Mathematics, 2014, v. 58, no. 1, p. 197-212 How to cite?
Journal: Science China. Mathematics 
Abstract: We study symmetric tensor spaces and cones arising from polynomial optimization and physical sciences. We prove a decomposition invariance theorem for linear operators over the symmetric tensor space, which leads to several other interesting properties in symmetric tensor spaces. We then consider the positive semidefiniteness of linear operators which deduces the convexity of the Frobenius norm function of a symmetric tensor. Furthermore, we characterize the symmetric positive semidefinite tensor (SDT) cone by employing the properties of linear operators, design some face structures of its dual cone, and analyze its relationship to many other tensor cones. In particular, we show that the cone is self-dual if and only if the polynomial is quadratic, give specific characterizations of tensors that are in the primal cone but not in the dual for higher order cases, and develop a complete relationship map among the tensor cones appeared in the literature.
URI: http://hdl.handle.net/10397/16662
ISSN: 1674-7283
DOI: 10.1007/s11425-014-4930-z
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