# Rings and Fields (Oxford Science Publications) by Graham Ellis

By Graham Ellis

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Intégration: Chapitres 7 et 8

Intégration, Chapitres 7 et 8Les Éléments de mathématique de Nicolas BOURBAKI ont pour objet une présentation rigoureuse, systématique et sans prérequis des mathématiques depuis leurs fondements. Ce quantity du Livre d’Intégration, sixième Livre du traité, traite de l’intégration sur les groupes localement compacts et de ses functions.

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We have that c−1 (c(T )) = T and c(c−1 (U )) = U ; c(T ) is permutable with T , and commutes with any S ∈ L(H) which is permutable with T . 2 Definition. The operator c(T ) is the Cayley transform of T , and c−1 (U ) is the inverse Cayley transform of U . If E is a spectral projection of c(T ) corresponding to a Borel subset A of T, then E is permutable with T ; in fact, EU is a unitary on EH, and it is easily veriﬁed that c−1 (EU ), which is a self-adjoint operator on EH, agrees with T |EH . Furthermore, E commutes with any bounded operator which is permutable with T .

Let S, T ∈ L(H) with S ∗ S ≤ T ∗ T . Then there is a unique W ∈ L(H) with W ∗ W ≤ QT (hence W ≤ 1), and S = W T . If R ∈ L(H) commutes with S, T , and T ∗ , then RW = W R. Proof: W is deﬁned on R(T ) by W (T ξ) = Sξ (W is well deﬁned since Sξ ≤ T ξ for all ξ). W extends to an operator on QT H by continuity; set ∗ ∗ ∗ W = 0 on Q⊥ T H. Then W W ≤ QT and S = W T . 6 The Spectral Theorem 23 and thus RW η = W Rη for η ∈ QT H. Since R(T T ∗ ) = (T T ∗ )R, RQT = QT R, and thus R leaves (I −QT )H invariant; thus if η ∈ Q⊥ T H, then W Rη = RW η = 0.

Let T be a densely deﬁned operator on H. Then T is symmetric if T ⊆ T ∗ , and T is self-adjoint if T = T ∗ . A symmetric operator is closable (its closure is also symmetric) and a selfadjoint operator is closed. 2 Examples. 7 Unbounded Operators 31 D0 = {f ∈ D2 : f (0) = f (1) = 0}. Then D0 is dense in L2 [0, 1], and Dk has codimension one in Dk+1 (k = 0, 1). Let Tk be deﬁned by Tk f = f with domain Dk (k = 0, 1, 2). Then each Tk is closed, T1 is self-adjoint, and T0∗ = T2 , so T0 is closed and symmetric, but not self-adjoint.