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Contents

Preface
1 Axiomatics for Functional Calculi
1.1 The Concept of Functional Calculus . . . .
1.2 An Abstract Framework . . . . . . . . . . .
1.2.1 The Extension Procedure . . . . . .
1.2.2 Properties of the Extended Calculus
1.2.3 Generators and Morphisms . . . . .
1.3 Meromorphic Functional Calculi . . . . . .
1.3.1 Rational Functions . . . . . . . . . .
1.3.2 An Abstract Composition Rule . . .
1.4 Multiplication Operators . . . . . . . . . . .
1.5 Concluding Remarks . . . . . . . . . . . . .
1.6 Comments . . . . . . . . . . . . . . . . . . .


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2 The Functional Calculus for Sectorial Operators
2.1 Sectorial Operators . . . . . . . . . . . . . . . . . . . . . .

2.1.1 Examples . . . . . . . . . . . . . . . . . . . . . . .
2.1.2 Sectorial Approximation . . . . . . . . . . . . . . .
2.2 Spaces of Holomorphic Functions . . . . . . . . . . . . . .
2.3 The Natural Functional Calculus . . . . . . . . . . . . . .
2.3.1 Primary Functional Calculus via Cauchy Integrals
2.3.2 The Natural Functional Calculus . . . . . . . . . .
2.3.3 Functions of Polynomial Growth . . . . . . . . . .
2.3.4 Injective Operators . . . . . . . . . . . . . . . . . .
2.4 The Composition Rule . . . . . . . . . . . . . . . . . . . .
2.5 Extensions According to Spectral Conditions . . . . . . .
2.5.1 Invertible Operators . . . . . . . . . . . . . . . . .
2.5.2 Bounded Operators . . . . . . . . . . . . . . . . .
2.5.3 Bounded and Invertible Operators . . . . . . . . .
2.6 Miscellanies . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6.1 Adjoints . . . . . . . . . . . . . . . . . . . . . . . .
2.6.2 Restrictions . . . . . . . . . . . . . . . . . . . . . .

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www.pdfgrip.com

vi

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3 Fractional Powers and Semigroups
3.1 Fractional Powers with Positive Real Part . . .
3.2 Fractional Powers with Arbitrary Real Part . .
3.3 The Phillips Calculus for Semigroup Generators
3.4 Holomorphic Semigroups . . . . . . . . . . . . .
3.5 The Logarithm and the Imaginary Powers . . .
3.6 Comments . . . . . . . . . . . . . . . . . . . . .

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4 Strip-type Operators and the Logarithm
4.1 Strip-type Operators . . . . . . . . . . . . . . . .
4.2 The Natural Functional Calculus . . . . . . . . .
4.3 The Spectral Height of the Logarithm . . . . . .
4.4 Monniaux’s Theorem and the Inversion Problem
4.5 A Counterexample . . . . . . . . . . . . . . . . .
4.6 Comments . . . . . . . . . . . . . . . . . . . . . .

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5 The Boundedness of the H ∞ -calculus

5.1 Convergence Lemma . . . . . . . . . . . . . . . . . . .
5.1.1 Convergence Lemma for Sectorial Operators. .
5.1.2 Convergence Lemma for Strip-type Operators.
5.2 A Fundamental Approximation Technique . . . . . . .
5.3 Equivalent Descriptions and Uniqueness . . . . . . . .
5.3.1 Subspaces . . . . . . . . . . . . . . . . . . . . .
5.3.2 Adjoints . . . . . . . . . . . . . . . . . . . . . .
5.3.3 Logarithms . . . . . . . . . . . . . . . . . . . .
5.3.4 Boundedness on Subalgebras of H ∞ . . . . . .
5.3.5 Uniqueness . . . . . . . . . . . . . . . . . . . .
5.4 The Minimal Angle . . . . . . . . . . . . . . . . . . . .
5.5 Perturbation Results . . . . . . . . . . . . . . . . . . .
5.5.1 Resolvent Growth Conditions . . . . . . . . . .
5.5.2 A Theorem of Pră
uss and Sohr . . . . . . . . . .
5.6 A Characterisation . . . . . . . . . . . . . . . . . . . .
5.7 Comments . . . . . . . . . . . . . . . . . . . . . . . . .

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2.7

2.8

2.6.3 Sectorial Approximation . . . . .
2.6.4 Boundedness . . . . . . . . . . .
The Spectral Mapping Theorem . . . . .
2.7.1 The Spectral Inclusion Theorem
2.7.2 The Spectral Mapping Theorem
Comments . . . . . . . . . . . . . . . . .

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