Linear Preserver Problems On Tensor Spaces Of Matrices

The study of invariants plays an important role in pure mathematics. Preserver problems are to characterize maps which preserve certain invariants on a given mathematical structure. Preserver problems are viewed as frontier and core research area in matrix theory. The linear preserver problems of multiple invariants have been studied by many scholars and generalized from many perspectives. At the International Conference on Matrix and Operator Theory in 2012, Professor Chi-Kwong Li, the vice president of International Linear Algebra Society, presented the framework of linear preserver problems on tensor spaces arising in quantum information science. In particular, he presented the open problems on linear maps which preserve ranks(more generally, rank-1) on tensor spaces of matrices. For such problems, one can expect to get more general forms of maps by restricting the range of invariants to the set of pure tensors and reducing the conditions of maps. But meanwhile, the relative study becomes difficult.The thesis focuses on the study of preserver problems on tensor spaces of matrices.The contents of this thesis are as follows.(1) To study the linear maps which preserve the ranks of tensor products of matrices. Example is shown to indicate that the linear maps, which preserve ranks of tensor products of matrices, do not preserve ranks of matrices in general. The canonical maps are introduced on tensor spaces of matrices. The basic properties of canonical maps are studied. Characterize the form of linear maps which preserve ranks of tensor products of matrices, and solve an open problem presented by Professor Chi-Kwong Li.(2) To study the linear maps which preserve tensor products of rank-1 Hermitian matrices. The basic properties of canonical maps on Hermitian are studied. The ascending chain of sets containing rank-1 Hermitian pure tensor is constructed. A class of linear maps which is determined by canonical maps are obtained. We also characterize the form of injective linear maps which preserve tensor products of rank-1 Hermitian matrices, and illustrate necessity of the injectivity assumption.(3) To study the linear maps which preserve idempotents of tensor products of(Hermitian) matrices. The ascending chain of sets containing idempotent Hermitian pure tensor is constructed. By the method of extension and restriction of maps, the linear maps which preserve idempotents of tensor products of(Hermitian) matrices are characterized.As applications, the forms of linear maps which preserve tripotents, M-P inverses, group inverses of tensor products of(Hermitian) matrices are characterized.The thesis contains the generalization of two central issues, rank preserver and idempotence preserver, in the classical linear preserver problems. The results enrich the existing theories of linear preserver problems on the matrix tensor product space.