Functional Analysis

Analyse mathématique IV: Intégration et théorie spectrale, - download pdf or read online

By Roger Godement

ISBN-10: 3540438416

ISBN-13: 9783540438410

Ce 4?me quantity de l'ouvrage Analyse math?matique initiera le lecteur ? l'analyse fonctionnelle (int?gration, espaces de Hilbert, examine harmonique en th?orie des groupes) et aux m?thodes de los angeles th?orie des fonctions modulaires (s?ries L et theta, fonctions elliptiques, utilization de l'alg?bre de Lie de SL2). Tout comme pour les volumes 1 ? three, on reconna?tra ici encore, le type inimitable de l'auteur et pas seulement par son refus de l'ecriture condens?e en utilization dans de nombreux manuels. Mariant judicieusement les math?matiques dites 'modernes' et' classiques', los angeles premi?re partie (Int?gration) est d'utilit? universelle tandis que los angeles seconde oriente le lecteur vers un domaine de recherche sp?cialis? et tr?s actif, avec de vastes g?n?ralisations possibles.

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Extra resources for Analyse mathématique IV: Intégration et théorie spectrale, analyse harmonique, le jardin des délices modulaires

Example text

On γ . 14 says that ( f /ϕ )i ∈ H 2 (D) whenever f ∈ N. Thus, since we are assuming that ΘN ≡ 1 (note the discussion in the second paragraph of the proof), it must be the case that ϕi is D-outer. 10 that a(1/z) ∈ N0+ (De ). e. 5) The first summand is integrable on T. 2) to see this. The second is integrable since |ϕe |2 w is integrable. Thus by standard Nevanlinna theory [32, Ch. 1) exists. 5. e. on E. ϕi We are now at the crux of the proof. e. on E . F To show N = N1 , we begin with the following claim.

If N is not H ∞ (C \ γ )-invariant, then the parameters Θ, E, and ρ are unique in the sense that if Θ j , E j , ρ j , j = 1, 2, represent N, then Θ1 = eit Θ2 , m(E1 ∆E2 ) = 0, and ρ1 χE1 = ρ2 χE2 almost everywhere. 2. If H ∞ (C \ γ )N ⊂ N, then Θ is unique up to a unimodular constant and m(E) = 0. Proof. The parameter Θ is the greatest common C \ γ -inner divisor of the functions in N and thus is unique up to a unimodular constant. 3) and m(E1 \ E2 ) > 0. Let Nρ j ,E j , j = 1, 2, denote N represented by ρ j , E j .

The above proposition implies that functions in the unit ball of H p (Ω) or E p (Ω) form a normal family on Ω. 1 Statement of the main result For a general domain Ω ⊂ C, a subspace1 N ⊂ H 2 (Ω) (not necessarily invariant) is said to be nearly invariant if there is some λ0 ∈ Ω such that whenever f ∈ N and f (λ0 ) = 0, then f ∈ N. z − λ0 The following useful proposition, found in [7, Prop. 1], says that we need not single out a particular λ0 . 1. If N is a subspace of H 2 (Ω), then the following are equivalent.

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Analyse mathématique IV: Intégration et théorie spectrale, analyse harmonique, le jardin des délices modulaires by Roger Godement

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