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.

The aim of this book is to provide an introduction to current advances of preserver problems on matrices and present the basic methods of studying preserver problems.In order to ensure that the content is self-contained,we took some conclusions and their proofs from other books or papers,which will be convenient to the readers.It is also intended to offer a collectio a of references on preserver problems on matrices.

This book is especially appropriate for advanced undergraduates majored in mathematics,and also intended for graduate students and research level mathematicians.It is hoped that the book will be suitab e for postgraduate to use or as a reference.The readers are required to familiarize with matrix methods and to comprehend some basic concepts in algebra,such as linear space,(semi-)group,(semi-)ring,homomorphism,isomorphism,etc.

Preface

Notation

Chapter 1 Introduction to preserver problems

 1.1 Definitions

 1.2 Typical categories of preserver problems

 1.3 Techniques of Preserver Problems

 1.4 Organization

Chapter 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 2

Chapter 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 3

Chapter 4 Additive preservers of rank equivalence and applications

 4.1 Main results

 4.2 Proofs of main results

 4.3 Remarks on Chapter 4

Chapter 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 5

Chapter 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 6

Chapter 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 7

Chapter 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 8

Chapter 9 Linear preservers of involutory matrices

 9.1 Main results

 9.2 Proofs of main results

 9.3 Remarks on Chapter 9

Chapter 10 Additive adjoint preservers

 10.1 Main results

 10.2 Proofs of main results

 10.3 Remarks on Chapter 10

Chapter 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 11

Appendix A Fitting decomposition of a matrix

Appendix B Canonical form of a rank-additive matrix pair

Appendix C Canonical form of a matrix triple

Appendix D Adjoint matrices

Bibliography

Index