矩阵空间的保持问题
2007-6
科学
张显
《矩阵空间的保持问题(英文版)》介绍矩阵论的一个研究领域----保持问题。主要内容包括最小秩保持、秩加性保持、秩等价保持、平方零保持、k-幂保持、k-幂等保持、矩阵(广义)逆保持、对合保持、伴随保持、乘积保持等。
PrefaceNotationChapter 1 Introduction to preserver problems 1.1 Definitions 1.2 Typical categories of preserver problems 1.3 Techniques of Preserver Problems 1.4 OrganizationChapter 2 Preservers of the smallest nonzero rank 2.1 Additive rank-1 preservers between spaces of rectangular matrices 2.2 Additive rank-1 preservers between spaces of symmetric matrices 2.3 Additive rank-1 preservers between spaces of Hermitian matrices 2.4 Additive rank-1 preservers from a space of triangular matrices to a space of square matrices 2.5 Additive rank-2 preservers between spaces of alternate matrices 2.6 Invertible linear rank-1 preservers on spaces of trace zero matrices 2.7 Invertibility (characteristic polynomial,determinant) preservers 2.8 Remarks on Chapter 2Chapter 3 Additive rank-additivity preservers and applications 3.1 Additive rank-additivity preservers on spaces of rectangular matrices 3.2 Additive rank-additivity preservers on spaces of symmetric matrices 3.3 Additive rank-additivity preservers on spaces of Hermitian matrices 3.4 Additive rank-additivity preservers on spaces of alternate matrices 3.5 Applications of additive rank-additivity preservers 3.6 Remarks on Chapter 3Chapter 4 Additive preservers of rank equivalence and applications 4.1 Main results 4.2 Proofs of main results 4.3 Remarks on Chapter 4Chapter 5 Linear square-zero preservers on spaces of square matrices 5.1 Main results 5.2 Proofs of main results 5.3 Remarks on Chapter 5Chapter 6 Preservers of k-power/k-potent 6.1 Additive preservers of idempotence between spaces of square matrices 6.2 Injective additive preservers of idempotence on spaces of triangular matrices 6.3 Linear preservers of idempotence between spaces of symmetric matrices 6.4 Several problems related to k-power and k-potent preservers 6.5 Preservers of k-power 6.6 Preservers of k-potent matrices 6.7 Jordan homomorphisms between of rings of square matrices 6.8 Preservers of idempotent relation 6.9 Remarks on Chapter 6Chapter 7 Preservers of inverses of matrices 7.1 Linear inverse preservers between spaces of matrices 7.2 Additive inverse preservers between spaces of matrices 7.3 Remarks on Chapter 7Chapter 8 Preservers of generalized inverses of matrices 8.1 Additive preservers of M-P inverses of matrices over fields 8.2 Additive preservers of {1,2}-inverses of matrices over fields 8.3 Additive preservers of group inverses of matrices over fields 8.4 Strong linear preservers of Moore-Penrose inverses of matrices over the two-element Boolean algebra 8.5 Covariant operator pairs on M-P Inverses of matrices between real finite dimensional division algebras 8.6 Remarks on Chapter 8Chapter 9 Linear preservers of involutory matrices 9.1 Main results 9.2 Proofs of main results 9.3 Remarks on Chapter 9Chapter 10 Additive adjoint preservers 10.1 Main results 10.2 Proofs of main results 10.3 Remarks on Chapter 10Chapter 11 Preservers of multiplication 11.1 Homomorphisms from one general linear group to another 11.2 Homomorphisms from a multiplicative matrix semigroup to another 11.3 Automorphisms of multiplicative semigroups of upper triangular matrices 11.4 Remarks on Chapter 11Appendix A Fitting decomposition of a matrixAppendix B Canonical form of a rank-additive matrix pairAppendix C Canonical form of a matrix tripleAppendix D Adjoint matricesBibliographyIndex
《矩阵空间的保持问题(英文版)》：One of the very active research areas in matrix theory is the study of preserver problems,including linear preserver problems,additive preserver problems and multiplicative preserver problems,which concern the classification of maps on matrices or operators that preserve certain special properties.
