Haase The Functional Calculus for Sectorial Operators

1. Auflage 2006
ISBN: 978-3-7643-7698-7
Verlag: Springer
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 394 Seiten

This book contains a systematic and partly axiomatic treatment of the holomorphic functional calculus for unbounded sectorial operators. The account is generic so that it can be used to construct and interrelate holomorphic functional calculi for other types of unbounded operators. Particularly, an elegant unified approach to holomorphic semigroups is obtained. The last chapter describes applications to PDE, evolution equations and approximation theory as well as the connection with harmonic analysis.

Autoren/Hrsg.

Haase, Markus

Inhaltsverzeichnis

1;1 Axiomatics for Functional Calculi;14
1.1;1.1 The Concept of Functional Calculus;14
1.2;1.2 An Abstract Framework;16
1.3;1.3 Meromorphic Functional Calculi;22
1.4;1.4 Multiplication Operators;26
1.5;1.5 Concluding Remarks;28
1.6;1.6 Comments;29
2;2 The Functional Calculus for Sectorial Operators;31
2.1;2.1 Sectorial Operators;31
2.2;2.2 Spaces of Holomorphic Functions;38
2.3;2.3 The Natural Functional Calculus;42
2.4;2.4 The Composition Rule;53
2.5;2.5 Extensions According to Spectral Conditions;57
2.6;2.6 Miscellanies;60
2.7;2.7 The Spectral Mapping Theorem;65
2.8;2.8 Comments;69
3;3 Fractional Powers and Semigroups;73
3.1;3.1 Fractional Powers with Positive Real Part;73
3.2;3.2 Fractional Powers with Arbitrary Real Part;82
3.3;3.3 The Phillips Calculus for Semigroup Generators;85
3.4;3.4 Holomorphic Semigroups;88
3.5;3.5 The Logarithm and the Imaginary Powers;93
3.6;3.6 Comments;100
4;4 Strip-type Operators and the Logarithm;102
4.1;4.1 Strip-type Operators;102
4.2;4.2 The Natural Functional Calculus;104
4.3;4.3 The Spectral Height of the Logarithm;109
4.4;4.4 Monniaux’s Theorem and the Inversion Problem;111
4.5;4.5 A Counterexample;112
4.6;4.6 Comments;115
5;5 The Boundedness of the Calculus;116
5.1;5.1 Convergence Lemma;116
5.2;5.2 A Fundamental Approximation Technique;119
5.3;5.3 Equivalent Descriptions and Uniqueness;122
5.4;5.4 The Minimal Angle;128
5.5;5.5 Perturbation Results;130
5.6;5.6 A Characterisation;138
5.7;5.7 Comments;138
6;6 Interpolation Spaces;141
6.1;6.1 Real Interpolation Spaces;141
6.2;6.2 Characterisations;144
6.3;6.3 Extrapolation Spaces;152
6.4;6.4 Homogeneous Interpolation;159
6.5;6.5 More Characterisations and Dore’s Theorem;163
6.6;6.6 Fractional Powers as Intermediate Spaces;167
6.7;6.7 Characterising Growth Conditions;174
6.8;6.8 Comments;178
7;7 The Functional Calculus on Hilbert Spaces;180
7.1;7.1 Numerical Range Conditions;182
7.2;7.2 Group Generators on Hilbert Spaces;194
7.3;7.3 Similarity Theorems for Sectorial Operators;203
7.4;7.4 Cosine Function Generators;217
7.5;7.5 Comments;221
8;8 Differential Operators;228
8.1;8.1 Elliptic Operators:;230
8.2;Theory;230
8.3;8.2 Elliptic Operators:;236
8.4;Theory;236
8.5;8.3 The Laplace Operator;240
8.6;8.4 The Derivative on the Line;246
8.7;8.5 The Derivative on a Finite Interval;249
8.8;8.6 Comments;256
9;9 Mixed Topics;260
9.1;9.1 Operators Without Bounded;260
9.2;Calculus;260
9.3;9.2 Rational Approximation Schemes;265
9.4;9.3 Maximal Regularity;276
10;A Linear Operators;288
10.1;A.1 The Algebra of Multi-valued Operators;288
10.2;A.2 Resolvents;291
10.3;A.3 The Spectral Mapping Theorem for the Resolvent;295
10.4;A.4 Adjoints;297
10.5;A.5 Convergence of Operators;299
10.6;A.6 Polynomials and Rational Functions of an Operator;301
10.7;A.7 Injective Operators;304
10.8;A.8 Semigroups and Generators;306
10.9;References;311
11;B Interpolation Spaces;312
11.1;B.1 Interpolation Couples;312
11.2;B.2 Real Interpolation by the K-Method;314
11.3;B.3 Complex Interpolation;319
11.4;References;322
12;C Operator Theory on Hilbert Spaces;323
12.1;C.1 Sesquilinear Forms;323
12.2;C.2 Adjoint Operators;325
12.3;C.3 The Numerical Range;328
12.4;C.4 Symmetric Operators;329
12.5;C.5 Equivalent Scalar;331
12.6;Products;331
12.7;and;331
12.8;the;331
12.9;Lax–Milgram;331
12.10;Theorem;331
12.11;C.6 Weak Integration;333
12.12;C.7 Accretive Operators;335
12.13;C.8 The Theorems of Plancherel and Gearhart;337
12.14;References;337
13;D The Spectral Theorem;339
13.1;D.1 Multiplication Operators;339
13.2;D.2 Commutative;341
13.3;Algebras. The Cyclic Case;341
13.4;D.3 Commutative;343
13.5;Algebras. The General Case;343
13.6;D.4 The Spectral Theorem: Bounded Normal;345
13.7;Operators;345
13.8;D.5 The Spectral Theorem: Unbounded Self-adjoint;346
13.9;Operators;346
13.10;D.6 The Functional Calculus;347
13.11;References;348
14;E Fourier Multipliers;349
14.1;E.1 The Fourier Transform on the Schwartz Space;349
14.2;E.2 Tempered Distributions;351
14.3;E.3 Convolution;353
14.4;E.4 Bounded Fourier Multiplier Operators;354
14.5;E.5 Some Pseudo-singular Multipliers;357
14.6;E.6 The Hilbert Transform and UMD Spaces;360
14.7;E.7 R-Boundedness and Weis’ Theorem;362
14.8;References;364
15;F Approximation by Rational Functions;365