Théorie spectrale et analyses complexes 44847

Extrait / Introduction

Extrait / Introduction :

cours sur la théorie spectrale et les complexes analyiques pour les étudiants inscris en master de mathématiques appliqués. These notes a r e issued from lectures given by the author a t the "Collkge de France" in 1971, the purpose of which was an exposition of complex analysis in Cn based on spectral theory. Such an approach leads to global theorems in connection with holomorphic convexity, approximation problems or ideals of holomorphic functions, and makes possible the introduction of growth conditions. It is easy to apply the holomorphic functional calculus of Banach algebras to polynomial approximation of holomorphic functions on a neighbourhood of a polynomially convex compact set K in C", by proving that K is the joint spectrum of the coordinates in the closed subalgebra generated by the polynomials i n e ( K ) , This method leads to the so-called Oka - Weil theorem. We remark that polynomial convexity is equivalent to the existence of a family (p, ) of polynomials such that

Plan

Plan :

CONTENTS INTRODUCTION ........................................... LIST OF SYMBOLS ........................................ CHAPTER I . . ALGEBRAS OF HOLOMORPHIC FUNCTIONS WITH RESTRICTED GROWTH 1 . 1 . Basic definitions ....................................... 1 . 2 . Weight functions ........................................ 1 . 3 . Elementary properties .................................. 1 . 4 . Regularization of weight functions ........................ 1 . 5 . Inductive limits ......................................... Notes .................................................. CHAPTERI1.- BOUNDEDNESS ANDPOLYNORMEDVECTOR SPACES 2 . 1 . Polynormed vector spaces ............................... 2.2. Convex bounded structures ............................... 2.3. Completeness ........................................... 2.4. Closure and density ..................................... 2 . 5 . Algebras and ideals ..................................... Notes .................................................. CHAPTER I11 . - SPECTRAL THEORY O F b-ALGEBRAS 3 . 1 . Spectrum of elements in a Banach algebra ................. 3.2. Spectral sets ........................................... 3 . 3 . Spectral functions ...................................... 3.4. The holomorphic functional calculus 3.5. Spectral theory modulo a b-ideal ....................... .......................... Notes .................................................. V x i 1 2 3 5 8 11 12 14 16 18 18 21 23 25 28 30 37 39 CHAPTER1V.- SPECTRALTHEOREMSANDHOLOMORPHIC CONVEXITY 4.1. Preliminaries ........................................... 41 4 . 2 . Spectral s e t s for z in O(S) .............................. 43 4 . 3 . Spectral functions for z in o(&) .......................... 47 4.4. Plurisubharmonic regularization ........................... 49 4.5. Domains of holomorphy .................................... 51 4.6. Bounded multiplicative linear forms ........................ 52 Notes ................................................... 53

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