The Research About Generalized Hilbert Tensors And Its Eigenvalues

The main purposes of this thesis focus on discussing the properties of infinite and finite dimensional generalized Hilbert tensors and extending the concept of Z1-eigenvalue from finite-dimensional tensors to infinite-dimensional tensors,and discussing the upper bound of such eigenvalues for infinite-dimensional generalized Hilbert tensorsFirstly,we give a introduction which mainly introduces the history and development of higher order tensor eigenvalues,Hilbert tensors,higher older Markov chains,transition probabil-ity tensors,Hilbert inequalities and Z1-eigenvalue,and then we expound the basic concepts and research status.Secondly,We mainly introduce the concept of Z1-eigenvalue of an m-order infinite-dimensional generalized Hilbert tensor H?? and achieve the upper bounds of Z1-eigenvalue,we also obtain the upper bounds of norm of the positively homogeneous operators which are defined by H??with ?>0Finally,we show the upper bounds of H-eigenvalue and Z-eigenvalue of an m-order(n+1)-dimensional generalized Hilbert tensor H?n when ??1.Then,we show the upper bound of its Z1-eigenvalue.In particular,when ?=1,it is a Hilbert tensor and we can attain the upper bound of its Z1-eigenvalue.