ALGEBRAIC MULTIPLICITY OF EIGENVALUES OF LINEAR OPERATORS
CONTENTS I FINITE-DIMENSIONAL CLASSIC SPECTRAL THEORY PART I: SUMMARY BASIC CONCEPTS THE JORDAN THEOREM THE JORDAN CANONICAL FORM THE REAL CANONICAL FORM AN EXAMPLE THE COMPLEX JORDAN FORM OF A THE REAL JORDAN FORM OF A EXERCISES COMMENTS ON CHAPTER 1 NORM OF A LINEAR OPERATOR INTRODUCTION TO OPERATOR CALCULUS RESOLVENT OPERATOR DUNFORD’S INTEGRAL FORMULA THE SPECTRAL MAPPING THEOREM THE EXPONENTIAL MATRIX COMMENTS ON CHAPTER 2 ESTIMATING THE INVERSE OF A MATRIX VECTOR-VALUED LAURENT SERIES THE EIGENVALUES ARE POLES OF THE RESOLVENT SPECTRAL PROJECTIONS COMMENTS ON CHAPTER 3 II ALGEBRAIC MULTIPLICITIES PART II: SUMMARY MOTIVATING THE CONCEPT OF TRANSVERSALITY THE CONCEPT OF TRANSVERSAL EIGENVALUE ALGEBRAIC EIGENVALUES AND TRANSVERSALIZATION PERTURBATION FROM SIMPLE EIGENVALUES COMMENTS ON CHAPTER 4 DERIVED FAMILIES AND FACTORIZATION CONNECTIONS BETWEEN ? AND µ COINCIDENCE OF THE MULTIPLICITIES ? AND µ A FORMULA FOR THE PARTIAL µ-MULTIPLICITIES REMOVABLE SINGULARITIES THE PRODUCT FORMULA PERTURBATION FROM SIMPLE EIGENVALUES REVISITED COMMENTS ON CHAPTER 5 SIMILARITY OF RANK-ONE PROJECTIONS PROOF OF THEOREM 6 0 1 RELAXING THE REGULARITY REQUIREMENTS A GENERAL UNIQUENESS THEOREM APPLICATIONS CLASSICAL MULTIPLICITY FORMULAE COMMENTS ON CHAPTER 6 THE CONCEPT OF JORDAN CHAIN CANONICAL SETS AND ?-MULTIPLICITY INVARIANCE BY CONTINUOUS FAMILIES OF ISOMORPHISMS LOCAL SMITH FORM FOR C?MATRIX FAMILIES CANONICAL SETS AT TRANSVERSAL EIGENVALUES COINCIDENCE OF THE MULTIPLICITIES ? AND ? LABELING THE VECTORS OF THE CANONICAL SETS CHARACTERIZING THE EXISTENCE OF THE SMITH FORM TWO ILLUSTRATIVE EXAMPLES EXAMPLE 1 EXAMPLE 2 7 10 LOCAL EQUIVALENCE OF OPERATOR FAMILIES 7 11 EXERCISES 7 12 COMMENTS ON CHAPTER 7 ISOLATED EIGENVALUES THE STRUCTURE OF THE SPECTRUM CLASSIC ALGEBRAIC MULTIPLICITY STABILITY OF THE COMPLEX ALGEBRAIC MULTIPLICITY LOCAL STABILITY GLOBAL STABILITY HOMOTOPY INVARIANCE COMMENTS ON CHAPTER 8 THE TRACE OPERATOR CONCEPT OF TRACE AND BASIC PROPERTIES TRACES OF THE COE?CIENTS OF THE PRODUCT OF TWO FAMILIES THE MULTIPLICITY THROUGH A LOGARITHMIC RESIDUE HOLOMORPHIC AND CLASSICAL FAMILIES SPECTRAL PROJECTION COMMENTS ON CHAPTER 9 10 THE SPECTRAL THEOREM FOR MATRIX POLYNOMIALS 10 1 LINEARIZATION OF A MATRIX POLYNOMIAL 10 2 GENERALIZED JORDAN THEOREM 10 3 REMARKS ON SCALAR POLYNOMIALS 10 4 CONSTRUCTING A BASIS IN THE PHASE SPACE 10 5 EXERCISES 10 6 COMMENTS ON CHAPTER 10 11 FURTHER DEVELOPMENTS OF THE ALGEBRAIC MULTIPLICITY 11 1 GENERAL FREDHOLM OPERATOR FAMILIES 11 2 MEROMORPHIC FAMILIES 11 3 UNBOUNDED OPERATORS 11 4 NON-FREDHOLM OPERATORS III NONLINEAR SPECTRAL THEORY PART III: SUMMARY 12 NONLINEAR EIGENVALUES 12 1 BIFURCATION VALUES NONLINEAR EIGENVALUES 12 2 A SHORT INTRODUCTION TO THE TOPOLOGICAL DEGREE 12 2 1 EXISTENCE AND UNIQUENESS 12 2 2 COMPUTING THE DEGREE ANALYTIC CONSTRUCTION 12 2 3 THE ?XED POINT THEOREM OF BROUWER, SCHAUDER AND TYCHONO? 12 2 4 FURTHER PROPERTIES 12 3 THE ALGEBRAIC MULTIPLICITY AS AN INDICATOR OF THE CHANGE OF INDEX 12 4 CHARACTERIZATION OF NONLINEAR EIGENVALUES 12 5 COMMENTS ON CHAPTER 12