Preface ......................................................... 7
Chapter 1
Fractional powers of operators ................................. 11
Chapter 2
Interpolation spaces ........................................... 17
2.1 Spaces Dσp ................................................ 18
2.2 Definition of interpolation spaces S(p,θ,X;p,θ -
1,Y) ...................................................... 23
2.3 Complex interpolation space ............................... 25
2.4 Another definition of interpolation spaces; Real
interpolation space ....................................... 27
2.4.1 The K-method ....................................... 28
2.4.2 The trace method ................................... 30
2.4.3 The Reiteration Theorem ............................ 32
2.4.4 Some examples ...................................... 33
Chapter 3
Infinitesimal generators of semi-groups ........................ 35
3.1 Infinitesimal generators of bounded semi-groups ........... 35
3.2 Infinitesimal generators of bounded analytic semi-
groups .................................................... 36
Chapter 4
Scales of Banach Spaces ........................................ 39
4.1 Inductive Limits and Projective Limits of Sequences of
Banach Spaces ............................................. 40
4.2 Regular Spaces and Hyper-spaces ........................... 51
Chapter 5
Examples of scales of Banach spaces ............................ 63
Chapter 6
Sectorial Operators ............................................ 71
6.1 Examples of Sectorial Operators ........................... 73
Chapter 7
Applications ................................................... 79
Chapter 8
The abstract Cauchy problem .................................... 89
8.1 Examples and applications ................................. 90
Appendix A
Theory of distributions and the Fourier transform .............. 99
A.l Theory of distributions ................................... 99
A.2 The Fourier transform of rapidly decreasing functions .... 100
A.3 The Fourier transform of tempered distributions .......... 101
Bibliography .................................................. 103
Streszczenie .................................................. 107
Резюме ........................................................ 107
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