Fredholm Theory in Banach Spaces

"In this Tract, Dr. Ruston presents analogues for operators on a Banach space of Fredholm's solution of integral equations of the second kind. Much of the presentation is based on research carried out over the last forty years and has never appeared in book form before."--BOOK JACKET.

Preface

ix

1 Introduction

1

(16)

1.1 The classical Fredholm theory

1

(4)

1.2 Summary

5

(2)

1.3 The complexification of real normed spaces

7

(10)

2 Asymptotically quasi-compact operators and the Riesz theory

17

(25)

2.1 Asymptotically quasi-compact operators

17

(6)

2.2 Null elements and the null space

23

(5)

2.3 Nucleus elements and the nucleus space

28

(5)

2.4 The Riesz decomposition of an operator on a Banach space

33

(9)

3 Tensor products of normed spaces

42

(36)

3.1 Introduction

42

(9)

3.2 Norms

51

(4)

3.3 The crossnorms γ and λ

55

(11)

3.4 The tensor product of a normed space and its conjugate

66

(12)

4 Fredholm theory for operators of finite rank

78

(40)

4.1 Tensors

78

(14)

4.2 Degenerate tensors

92

(12)

4.3 The Fredholm coefficients

104

(5)

4.4 The Fredholm formulae using determinants

109

(3)

4.5 The Fredholm theory

112

(6)

5 Operators with a Fredholm determinant

118

(17)

5.1 Fredholm formulae

118

(6)

5.2 Fredholm theory

124

(2)

5.3 Riesz theory

126

(1)

5.4 Riesz operators and Fredholm determinants

127

(8)

6 Fredholm formulae and the Riesz theory

135

(31)

6.1 Fredholm formulae

135

(3)

6.2 The Riesz theory

138

(16)

6.3 The Riesz decomposition

154

(3)

6.4 Alternative bases for the null space

157

(9)

7 Fredholm formulae for special classes of operators

166

(48)

7.1 The trace class of nuclear operators

166

(11)

7.2 The Lezanski-Sikorski theory

177

(17)

7.3 The Grothendieck theory

194

(10)

7.4 The Saphar theory

204

(10)

8 Notes and Comments

214

(36)

8.1 Introduction

214

(8)

8.2 Asymptotically quasi-compact operators and the Riesz theory

222

(4)

8.3 Tensor products of normed spaces

226

(8)

8.4 Fredholm theory for operators of finite rank

234

(1)

8.5 Operators with a Fredholm determinant

234

(7)

8.6 Fredholm formulae and the Riesz theory

241

(3)

8.7 Fredholm formulae for special classes of operators

244

(4)

8.8 Further developments

248

(2)

Bibliography

250

(35)

Index of notations

285

(3)

Subject index

288

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Fredholm Theory in Banach Spaces

Summary

Table of Contents

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Fredholm Theory in Banach Spaces

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