| Preface | p. ix |
| Prologue | p. 1 |
| Tapping away in an evening at Djursholm | p. 1 |
| What does an epsilon weigh? | p. 1 |
| Red wine at the Stock Exchange Club | p. 2 |
| Ice-cream in Madison | p. 2 |
| Exact equality | p. 3 |
| The deficiency of values | p. 5 |
| Zurich | p. 5 |
| Beautiful to look at, but... | p. 6 |
| The unbearable ease of using norms | p. 6 |
| Centenary Colloquium in Joensuu | p. 7 |
| Two basic tasks, stability first | p. 7 |
| And then accelerating the iteration | p. 9 |
| Factoring the resolvent | p. 10 |
| In the Hermann Weyl lecture hall | p. 11 |
| A quiet life in Warsaw | p. 12 |
| Finally, in Kirkkonummi | p. 12 |
| p. 15 |
| Resolvent | p. 15 |
| Cauchy-integral | p. 20 |
| p. 23 |
| Entire functions | p. 23 |
| Taylor coefficients | p. 24 |
| Meromorphic functions | p. 25 |
| The first main theorem | p. 30 |
| Cartan's identity | p. 31 |
| Order and type for meromorphic functions | p. 32 |
| Boutroux-Cartan lemma | p. 33 |
| Bound along a circle | p. 34 |
| Representation theorems | p. 36 |
| p. 37 |
| Analytic vector valued functions | p. 37 |
| Subharmonic functions | p. 37 |
| Meromorphic vector valued functions | p. 38 |
| Rational functions | p. 40 |
| When is the inverse also meromorphic | p. 42 |
| A simple estimate for matrices | p. 44 |
| p. 47 |
| A product form for matrices | p. 47 |
| Singular value decomposition | p. 50 |
| Basic inequalities for singular values and eigenvalues | p. 51 |
| The total logarithmic size of a matrix | p. 54 |
| Some basic properties of the total logarithmic size | p. 56 |
| Direct sum, Kronecker product and Hadamard product | p. 60 |
| p. 63 |
| The total logarithmic size is subharmonic | p. 63 |
| Behavior near poles | p. 65 |
| Introducing T[subscript 1] for matrix valued functions | p. 68 |
| Basic identity for inversion | p. 69 |
| Extension to trace class | p. 70 |
| How to work outside the trace class | p. 71 |
| p. 73 |
| Perturbation results | p. 73 |
| Special results for resolvents | p. 77 |
| Powers and their resolvents | p. 79 |
| Bounded characteristics | p. 83 |
| What if small perturbation means small in norm | p. 85 |
| p. 87 |
| Combining a scalar function with an operator | p. 87 |
| Representing F as G/g | p. 93 |
| Representations for the resolvent | p. 94 |
| Decay of spectral polynomials | p. 96 |
| Robust bounds for Krylov solvers | p. 98 |
| A bound for spectral projectors | p. 100 |
| p. 103 |
| Approximate polynomial degree of an analytic function | p. 103 |
| Some properties of the approximate polynomial degree | p. 106 |
| Approximate rational degree of a meromorphic function | p. 108 |
| Spijker's lemma | p. 112 |
| Power bounded operators and bounds for the Laurent coefficients | p. 114 |
| p. 117 |
| Growth of associated scalar functions | p. 117 |
| Locally algebraic and locally almost algebraic operators | p. 121 |
| p. 125 |
| Exceptional values | p. 125 |
| Simple asymptotics for resolvents of matrices | p. 126 |
| Eigenvalues and exceptional values | p. 128 |
| Deficient operators | p. 131 |
| Epilogue | p. 133 |
| Lecturing and typing in Toronto | p. 133 |
| Fishing and finishing in Karjalohja | p. 133 |
| Bibliography | p. 135 |
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