Some nonlinear spectral properties of higher order tensors

Abstract

The main purposes of this thesis focus on the nonlinear spectral properties of higher order tensor with the help of the spectral theory and fixed point theory of nonlinear positively homogeneous operator as well as the constrained minimization theory of homogeneous polynomial. The main contributions of this thesis are as follows. We obtain the Fredholm alternative theorems of the eigenvalue (included E-eigenvalue, H-eigenvalue, Z-eigenvalue) of a higher order tensor A. Some relationship between the Gelfand formula and the spectral radius are discussed for the spectra induced by such several classes of eigenvalues of a higher order tensor. This content is mainly based on the paper 5 in Underlying Papers. We show that the eigenvalue problem of a nonnegative tensor A can be viewed as the fixed point problem of the Edelstein Contraction with respect to Hilbert's projective metric. Then by means of the Edelstein Contraction Theorem, we deal with the existence and uniqueness of the positive eigenvalue-eigenvector of such a tensor, and give an iteration sequence for finding positive eigenvalue of such a tensor, i.e., a nonlinear version of the famous Krein-Rutman Theorem. This content is mainly based on the paper 2 in Underlying Papers. We introduce the concept of eigenvalue to the additively homogeneous mapping pairs (f, g), and establish existence and uniqueness of such a eigenvalue under the boundedness of some orbits of f, g in the Hilbert semi-norm. In particular, the nonlinear Perron-Frobenius property for nonnegative tensor pairs (A, B) is given without involving the calculation of the tensor inversion. Moreover, we also present the iteration methods for finding generalized H-eigenvalue of nonnegative tensor pairs (A, B). This content is mainly based on the paper 1 in Underlying Papers. We introduce the concepts of Pareto H-eigenvalue and Pareto Z-eigenvalue of higher order tensor for studying constrained minimization problem and show the necessary and sufficient conditions of such eigenvalues. We obtain that a symmetric tensor has at least one Pareto H-eigenvalue (Pareto Z-eigenvalue). What is more, the minimum Pareto H-eigenvalue (or Pareto Z-eigenvalue) of a symmetric tensor is exactly equal to the minimum value of constrained minimization problem of homogeneous polynomial deduced by such a tensor, which gives an alternative methods for solving the minimum value of constrained minimization problem. In particular, a symmetric tensor A is copositive if and only if every Pareto H-eigenvalue (Z.eigenvalue) of A is non-negative. This content is mainly based on the papers 3 and 4 in Underlying Papers.