The Research About Properties Of Generalized Hilbert Tensors And B Tensors

This thesis is mainly discussing the properties of infinite and finite dimensional gen-eralized Hilbert tensors and proving the boundedness of the solution set of tensor comple-mentarity problem with B tensor,such the specific bounds depend only on the structural properties of B tensor.The thesis consists of three chapters:In chapter 1,we give a introduction which mainly introduces the research status of various structured tensors and tensor complementarity problem,and the related basic conceptions.In chapter 2,first we introduce the concept of an m order n dimensional generalized Hilbert tensor Hn and show that its H spectral radius and its Z spectral radius are smaller than or equal to M（a）nm-1 and M（a）nm/2,respectively.here M（a）is a constant which only depends on a.In addition,if a>1,then infinite and finite dimensional generalized Hilbert tensors are both positive definite.Subsequently,for an m order infinite dimensional generalized Hilbert tensor H∞ with a>0,we show that H∞ defines a bounded,continuous and positively（m-1）-homogeneous operator from l1 into lp（1<p<∞）.Besides,we also obtain two upper bounds which on the norms of corresponding positively homogeneous operators.In chapter 3,our aim is to prove the boundedness of the solution set of tensor com-plementarity problem with B tensor,Moreover,the specific bounds only depend on the structural properties of B tensor.In order to achieve this goal,firstly,we prove that each B₀（B）tensor is（strictly）semi-positive.then,As to different operator norms of two pos-itively homogeneous operators which defined by B tensor,we give its strictly lower and upper bounds.Finally,Based on the above upper bounds,the strictly lower bound of solution set of tensor complementarity problem with B tensor are obtained.Besides,the upper bounds of spectral radius and E spectral radius are given for B（B₀）tensor,which depend only on the principal diagonal entries of B（B₀）tensor.