THE THEORY OF FRACTIONAL POWERS OF OPERATORS

Transcription

1 THE THEORY OF FRACTIONAL POWERS OF OPERATORS Celso MARTINEZ CARRACEDO and Miguel SANZ ALIX Departament de Matematica Aplicada Universitat de Valencia C/Dr. Moliner Burjassot Valencia Spain 2001 ELSEVIER Amsterdam - London - New York - Oxford - Paris - Shannon - Tokyo

2 Contents Introduction ix 1 Non-Negative Operators Definition and Basic Properties Sectorial Operators Examples of Non-Negative Operators m-accretive Operators Negative of Generators of Semigroups Product of Operators Multiplication Operators Generation of Sectorial Operators Komatsu's Counterexample Normal or Self-adjoint Sectorial Operators Non-Negative Operators in Locally Convex Spaces Non-Negative Operators which are Not Sectorial Notes on Chapter Differential Operators Operators of Riemann-Liouville and Weyl The Derivative Operator in R The Laplacian Operator Second-Order Elliptic Differential Operators Case 1: p = Case 2: 1 < p < oo Case 3: The Operator A c Case 4: p = The Laplacian in a Locally Convex Space of Distributions Notes on Chapter The Balakrishnan Operator Definition of Balakrishnan and Basic Properties Expressions of the Balakrishnan Operator when A is the Infinitesimal Generator of an Equibounded Co-Sernigroup Examples 69

3 vi CONTENTS 3.4 Notes on Chapter An Extension of the Hirsch Functional Calculus Classes of Functions Associated to Radon Measures Description of Different Classes of Functions Examples Uniqueness and Pointwise Convergence A Characterization of the Class T Functional Calculus Definition and Basic Properties Product Formula Stability under Composition Spectral Mapping Theorem Hirsch Functional Calculus in Locally Convex Spaces Integration in Locally Convex Spaces Remarks on the Hirsch Functional Calculus in Locally Convex Spaces Notes on Chapter Fractional Powers of Operators Definition of Fractional Power. Additivity Representations of the Fractional Powers Spectral Mapping Theorem Sectoriality of the Fractional Powers. Multiplicativity Semigroups Generated by Fractional Powers Fractional Powers of Operators in Locally Convex Spaces Notes on Chapter Domains, Uniqueness and the Cauchy Problem Domains of Fractional Powers Domains of the Fractional Powers when A Generates an Equibounded Co-Semigroup Conditions for Uniqueness Sufficient Conditions for Uniqueness Spectral and Analyticity Conditions : n-th Roots of a Non-Negative Operator The Second - Order Abstract Incomplete Cauchy Problem Results in Locally Convex Spaces Notes on Chapter Negative and Imaginary Powers Definitions and Basic Properties The Balakrishnan and Komatsu Operators Examples Limit Operators Related to the Imaginary Power Negative and Imaginary Powers on Locally Convex Spaces

4 Vll 7.6 Notes on Chapter The Dore-Venni Theorem Definitions and Notations Sectoriality and Boundedness of Exponential Type The Dore Venni Theorem r Sum of Closed Operators in UMD Spaces I? Maximal Regularity Notes on Chapter Functional Calculus for Co-groups The Mellin Transform Functional Calculus for Co-groups Analytic Generators. Imaginary Powers of the Product Imaginary Powers of the Sum of Operators Notes on Chapter Imaginary Powers on Hilbert Spaces Logarithms Relationship between Power Angle and Spectral Angle Bounded Functional Calculus Notes on Chapter Fractional Powers and Interpolation Spaces Introduction Interpolation Spaces. The Real Method Komatsu's Spaces Komatsu's Spaces and Real Interpolation Spaces Domains of Fractional Powers and the Komatsu Spaces Interpolation Spaces. The Complex Method Notes on Chapter Fractional Powers of some Differential Operators Imaginary Powers of Differential Polynomials in W (E n ) Imaginary Powers of Derivative Operators Imaginary Powers of the Negative of the Laplacian Riesz and Bessel Potentials Fractional Sobolev Spaces Notes on Chapter A Appendix 307 A.I Nets 307 A.2 Linear Operators 307 A.3 Functional Analysis 310 A.3.1 Banach Spaces 310 A.3.2 Banach Algebras 312 A.3.3 Hilbert Spaces 313

5 A.4 Measure and Integration ' 315 A.4.1 Lebesgue Spaces 315 A.4.2 The Radon-Riesz Representation Theorem 316 A.4.3 Calculus Facts 319 A.5 The Stone-Weierstrass Theorem 320 A.6 Vector-Valued Functions 321 A.6.1 Analytic Functions 321 A.6.2 Integration Theory for Vector-Valued Functions 323 A.7 Semigroups 326 A. 8 Convolution and Fourier Transforms 330 A.9 Harmonic Analysis 333 A.9.1 The Mikhlin Multiplier Theorem 333 A.9.2 The Riesz-Thorin Convexity Theorem 334 A. 10 Distributions 335 A.ll Sobolev Spaces 337 Notations 341 Bibliography 346 via