Multiparameter Eigenvalue Problems: Matrices and Compact Operators Volume 1

Front Cover......Page 1 Multiparameter Eigenvalue Problems......Page 4 Copyright Page......Page 5 Contents......Page 6 Preface......Page 10 Contents of Volume II......Page 12 PART I: PRELIMINARIES FROM LINEAR ALGEBRA......Page 14 1.1 Introduction......Page 16 1.2 Linear Maps......Page 19 1 3 Composite and Induced Maps......Page 21 1.4 Direct Sums......Page 23 1.5 Linear Dependence and Dimension......Page 25 1.6 Dimensions of Kernel and Image......Page 28 1.7 Further Dimensional Results......Page 29 1.8 Topologies......Page 31 1.9 Connectedness......Page 33 1.10 Semilinear Maps......Page 34 2.1 Multilinear Functions......Page 35 2.2 Bilinear Functions......Page 38 2.3 Bilinear Functions on a Single Space......Page 39 2.4 Bilinear Forms......Page 41 2.5 Sesquilinear Functions......Page 42 2.6 Sesquilinear Forms and Endoniorphisms......Page 45 2.7 The Zeros of Hermitian Forms......Page 46 2.8 Pairs of Hermitian Forms......Page 48 2.9 Three Hermitian Forms......Page 50 2.10 General Remarks on the Range of a Set of Forms......Page 52 3.1 Ascent and Descent......Page 55 3.2 The Case of Equal Ascent and Descent......Page 58 3.3 Eigensubspaces and Root Subspaces......Page 60 3.4 The Splitting Off of Root Subspaces......Page 61 3.5 The Finite-Dimensional Case......Page 62 3.6 Several Commuting Operators......Page 65 3.7 The Hermitian Case......Page 66 3.8 Orthogonality......Page 68 3.9 Some Modifications......Page 70 3.10 Reduction of Pairs of Hermitian Forms......Page 72 4.1 Introduction......Page 75 4.2 The Definition by Means of Functionals......Page 76 4.3 Bases and Dimension......Page 78 4.4 The Real and Complex Cases......Page 81 4.5 Subspaces......Page 82 4.6 Induced Homomorphisms......Page 84 4.7 Exactness Properties......Page 86 4.8 Universal Property......Page 88 4.9 Bilinear Forms and Tensor Products......Page 89 4.10 Products of Sesquilinear Forms......Page 91 5.1 Introduction......Page 94 5.2 The Hermitian Case......Page 95 5.3 Eigenvalues and Ranks......Page 97 5.4 Decomposition......Page 98 5.5 The Kronecker Sum and Product......Page 99 5.6 Kronecker Sums and Eigenvalues......Page 101 5.7 Separation of Variables......Page 103 5.8 The Tensor Product of Identical Factors......Page 104 5.9 Induced Maps of Symmetry Subspaces......Page 106 PART 2: MULTIPARAMETER PROBLEMS FOR MATRICES......Page 110 6.1 Introduction......Page 112 6.2 Determinantal Maps......Page 116 6.3 Singular Determinantal Maps in the Case k = 2......Page 118 6.4 Rectangular Arrays......Page 120 6.5 Definiteness Requirements......Page 122 6.6 Solutions for Rectangular Arrays......Page 123 6.7 Nonformal Determinantal Properties......Page 125 6.8 Eigenvalues for a Rectangular Array......Page 126 6.9 Decomposition......Page 127 7.1 Introduction......Page 130 7.2 The First Definiteness Condition and Its Consequences......Page 132 7.3 Orthogonality of Eigenvectors......Page 134 7.4 Stronger Definiteness Conditions......Page 136 7.5 Splitting of Multiple Eigenvalues......Page 138 7.6 Decomposable Orthogonal Eigenvectors......Page 142 7.7 A Connectedness Property......Page 144 7.8 The Main Result on Positive Definiteness......Page 147 7.9 The Eigenvector Expansion......Page 148 8.1 Introduction......Page 151 8.2 Equivalent Singularity Conditions......Page 152 8.3 An Algebraic Lemma......Page 153 8.4 The Inductive Argument......Page 155 8.5 Singularity and Decomposable Tensors......Page 158 8.7 Eigenvalues and Singularity......Page 159 9.1 Introduction......Page 162 9.2 Two by Two Arrays......Page 163 9.3 Two by Three Arrays......Page 165 9.4 General Square Arrays......Page 166 9.5 A Property of Convex Cones......Page 168 9.6 The Case of Several Cones......Page 174 9.7 Square Arrays of Hermitian Forms, Continued......Page 176 9.8 Rectangular Arrays of Hermitian Forms......Page 178 9.9 Relation between Definiteness Conditions I and II......Page 183 10.1 Introduction......Page 187 10.2 Eigenvalues and Eigensubspaces......Page 188 10.3 Eigenprojectors......Page 190 10.4 Existence of a Nonsingular Determinantal Map......Page 192 10.5 Completeness of the Eigenvectors......Page 194 10.6 The Eigenvector Expansion......Page 195 11.1 Introduction......Page 196 11.2 Notions from Hilbert Space Theory......Page 198 11.3 Discreteness of the Spectrum......Page 199 11.4 Truncated Problems......Page 202 11.5 Sequences of Truncations......Page 203 11.6 Convergence of the Eigenvalues......Page 204 11.7 Convergence of the Eigenvectors......Page 205 11.8 Introduction of Tensor Products......Page 207 11.9 Discussion of the Expansion......Page 212 11.10 A Special Case......Page 214 References......Page 218 Index......Page 222