The Functional Calculus for Sectorial Operators

by Haase, Markus

• ISBN13: 9783764376970
• ISBN10: 376437697X
• Format: Hardcover
• Copyright: 2006-08-14
• Publisher: Birkhauser

Summary

The present monograph deals with the functional calculus for unbounded operators in general and for sectorial operators in particular. Sectorial operators abound in the theory of evolution equations, especially those of parabolic type. They satisfy a certain resolvent condition that leads to a holomorphic functional calculus based on Cauchy-type integrals. Via an abstract extension procedure, this elementary functional calculus is then extended to a large class of (even meromorphic) functions. With this functional calculus at hand, the book elegantly covers holomorphic semigroups, fractional powers, and logarithms. Special attention is given to perturbation results and the connection with the theory of interpolation spaces. A chapter is devoted to the exciting interplay between numerical range conditions, similarity problems and functional calculus on Hilbert spaces. Two chapters describe applications, for example to elliptic operators, to numerical approximations of parabolic equations, and to the maximal regularity problem. This book is the first systematic account of a subject matter which lies in the intersection of operator theory, evolution equations, and harmonic analysis. It is an original and comprehensive exposition of the theory as a whole. Written in a clear style and optimally organised, it will prove useful for the advanced graduate as well as for the experienced researcher.

Table of Contents

Preface	xi
1 Axiomatics for Functional Calculi	1	(18)
1.1 The Concept of Functional Calculus	1	(2)
1.2 An Abstract Framework	3	(6)
1.2.1 The Extension Procedure	4	(1)
1.2.2 Properties of the Extended Calculus	5	(2)
1.2.3 Generators and Morphisms	7	(2)
1.3 Meromorphic Functional Calculi	9	(4)
1.3.1 Rational Functions	10	(2)
1.3.2 An Abstract Composition Rule	12	(1)
1.4 Multiplication Operators	13	(2)
1.5 Concluding Remarks	15	(1)
1.6 Comments	16	(3)
2 The Functional Calculus for Sectorial Operators	19	(42)
2.1 Sectorial Operators	19	(7)
2.1.1 Examples	24	(1)
2.1.2 Sectorial Approximation	25	(1)
2.2 Spaces of Holomorphic Functions	26	(4)
2.3 The Natural Functional Calculus	30	(11)
2.3.1 Primary Functional Calculus via Cauchy Integrals	30	(4)
2.3.2 The Natural Functional Calculus	34	(3)
2.3.3 Functions of Polynomial Growth	37	(2)
2.3.4 Injective Operators	39	(2)
2.4 The Composition Rule	41	(4)
2.5 Extensions According to Spectral Conditions	45	(3)
2.5.1 Invertible Operators	45	(1)
2.5.2 Bounded Operators	46	(1)
2.5.3 Bounded and Invertible Operators	47	(1)
2.6 Miscellanies	48	(5)
2.6.1 Adjoints	48	(2)
2.6.2 Restrictions	50	(1)
2.6.3 Sectorial Approximation	50	(2)
2.6.4 Boundedness	52	(1)
2.7 The Spectral Mapping Theorem	53	(4)
2.7.1 The Spectral Inclusion Theorem	53	(2)
2.7.2 The Spectral Mapping Theorem	55	(2)
2.8 Comments	57	(4)
3 Fractional Powers and Semigroups	61	(30)
3.1 Fractional Powers with Positive Real Part	61	(9)
3.2 Fractional Powers with Arbitrary Real Part	70	(3)
3.3 The Phillips Calculus for Semigroup Generators	73	(3)
3.4 Holomorphic Semigroups	76	(5)
3.5 The Logarithm and the Imaginary Powers	81	(7)
3.6 Comments	88	(3)
4 Strip-type Operators and the Logarithm	91	(14)
4.1 Strip-type Operators	91	(2)
4.2 The Natural Functional Calculus	93	(5)
4.3 The Spectral Height of the Logarithm	98	(2)
4.4 Monniaux's Theorem and the Inversion Problem	100	(1)
4.5 A Counterexample	101	(3)
4.6 Comments	104	(1)
5 The Boundedness of the H"-calculus	105	(26)
5.1 Convergence Lemma	105	(3)
5.1.1 Convergence Lemma for Sectorial Operators.	105	(2)
5.1.2 Convergence Lemma for Strip-type Operators.	107	(1)
5.2 A Fundamental Approximation Technique	108	(3)
5.3 Equivalent Descriptions and Uniqueness	111	(6)
5.3.1 Subspaces	112	(1)
5.3.2 Adjoint'	113	(1)
5.3.3 Logarithms	113	(1)
5.3.4 Boundedness on Subalgebras of Hinfinity	114	(2)
5.3.5 Uniqueness	116	(1)
5.4 The Minimal Angle	117	(2)
5.5 Perturbation Results	119	(8)
5.5.1 Resolvent Growth Conditions	119	(6)
5.5.2 A Theorem of Prüss and Sohr	125	(2)
5.6 A Characterisation	127	(1)
5.7 Comments	127	(4)
6 Interpolation Spaces	131	(40)
6.1 Real Interpolation Spaces	131	(3)
6.2 Characterisations	134	(8)
6.2.1 A First Characterisation	134	(5)
6.2.2 A Second Characterisation	139	(1)
6.2.3 Examples	140	(2)
6.3 Extrapolation Spaces	142	(7)
6.3.1 An Abstract Method	142	(2)
6.3.2 Extrapolation for Injective Sectorial Operators	144	(2)
6.3.3 The Homogeneous Fractional Domain Spaces	146	(3)
6.4 Homogeneous Interpolation	149	(4)
6.4.1 Some Intermediate Spaces	149	(3)
6.4.2 ...Are Actually Real Interpolation Spaces	152	(1)
6.5 More Characterisations and Dore's Theorem	153	(4)
6.5.1 A Third Characterisation (Injective Operators)	153	(2)
6.5.2 A Fourth Characterisation (Invertible Operators)	155	(1)
6.5.3 Dore's Theorem Revisited	156	(1)
6.6 Fractional Powers as Intermediate Spaces	157	(7)
6.6.1 Density of Fractional Domain Spaces	157	(1)
6.6.2 The Moment Inequality	158	(2)
6.6.3 Reiteration and Komatsu's Theorem	160	(2)
6.6.4 The Complex Interpolation Spaces and BIP	162	(2)
6.7 Characterising Growth Conditions	164	(4)
6.8 Comments	168	(3)
7 The Functional Calculus on Hilbert Spaces	171	(48)
7.1 Numerical Range Conditions	173	(12)
7.1.1 Accretive and w-accretive Operators	173	(2)
7.1.2 Normal Operators	175	(2)
7.1.3 Functional Calculus for m-accretive Operators	177	(3)
7.1.4 Mapping Theorems for the Numerical Range	180	(1)
7.1.5 The Crouzeix–Delyon Theorem	181	(4)
7.2 Group Generators on Hilbert Spaces	185	(9)
7.2.1 Liapunov's Direct Method for Groups	185	(3)
7.2.2 A Decomposition Theorem for Group Generators	188	(2)
7.2.3 A Characterisation of Group Generators	190	(4)
7.3 Similarity Theorems for Sectorial Operators	194	(14)
7.3.1 The Theorem of McIntosh	195	(2)
7.3.2 Interlude: Operators Defined by Sesquilinear Forms	197	(5)
7.3.3 Similarity Theorems	202	(3)
7.3.4 A Counterexample	205	(3)
7.4 Cosine Function Generators	208	(4)
7.5 Comments	212	(7)
8 Differential Operators	219	(32)
8.1 Elliptic Operators: L1-Theory	221	(6)
8.2 Elliptic Operators: Lp-Theory	227	(4)
8.3 The Laplace Operator	231	(6)
8.4 The Derivative on the Line	237	(3)
8.5 The Derivative on a Finite Interval	240	(7)
8.6 Comments	247	(4)
9 Mixed Topics	251	(28)
9.1 Operators Without Bounded Hinfinity-Calculus	251	(5)
9.1.1 Multiplication Operators for Schauder Bases	251	(2)
9.1.2 Interpolating Sequences	253	(1)
9.1.3 Two Examples	254	(2)
9.1.4 Comments	256	(1)
9.2 Rational Approximation Schemes	256	(11)
9.2.1 Time-Discretisation of First-Order Equations	257	(2)
9.2.2 Convergence for Smooth Initial Data	259	(2)
9.2.3 Stability	261	(4)
9.2.4 Comments	265	(2)
9.3 Maximal Regularity	267	(12)
9.3.1 The Inhomogeneous Cauchy Problem	267	(1)
9.3.2 Sums of Sectorial Operators	268	(5)
9.3.3 (Maximal) Regularity	273	(4)
9.3.4 Comments	277	(2)
A Linear Operators	279	(24)
A.1 The Algebra of Multi-valued Operators	279	(3)
A.2 Resolvents	282	(4)
A.3 The Spectral Mapping Theorem for the Resolvent	286	(2)
A.4 Adjoints	288	(2)
A.5 Convergence of Operators	290	(2)
A.6 Polynomials and Rational Functions of an Operator	292	(3)
A.7 Injective Operators	295	(2)
A.8 Semigroups and Generators	297	(6)
B Interpolation Spaces	303	(12)
B.1 Interpolation Couples	303	(2)
B.2 Real Interpolation by the K-Method	305	(5)
B.3 Complex Interpolation	310	(5)
C Operator Theory on Hilbert Spaces	315	(16)
C.1 Sesquilinear Forms	315	(2)
C.2 Adjoint Operators	317	(3)
C.3 The Numerical Range	320	(1)
C.4 Symmetric Operators	321	(2)
C.5 Equivalent Scalar Products and the Lax Milgram Theorem	323	(2)
C.6 Weak Integration	325	(2)
C.7 Accretive Operators	327	(2)
C.8 The Theorems of Plancherel and Gearhart	329	(2)
D The Spectral Theorem	331	(10)
D.1 Multiplication Operators	331	(2)
D.2 Commutative C*-Algebras. The Cyclic Case	333	(2)
D.3 Commutative C*-Algebras. The General Case	335	(2)
D.4 The Spectral Theorem: Bounded Normal Operators	337	(1)
D.5 The Spectral Theorem: Unbounded Self-adjoint Operators	338	(1)
D.6 The Functional Calculus	339	(2)
E Fourier Multipliers	341	(16)
E.1 The Fourier Transform on the Schwartz Space	341	(2)
E.2 Tempered Distributions	343	(2)
E.3 Convolution	345	(1)
E.4 Bounded Fourier Multiplier Operators	346	(3)
E.5 Some Pseudo-singular Multipliers	349	(3)
E.6 The Hilbert Transform and UMD Spaces	352	(2)
E.7 R-Boundedness and Weis' Theorem	354	(3)
F Approximation by Rational Functions	357	(4)
Bibliography	361	(16)
Index	377	(8)
Notation	385