Woo Young Lee Lecture Notes on Operator Theory

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CHAPTER 5. TOEPLITZ THEORY Proof. (i) Since by Parseval’s identity, ∑ | ̂f(n)| 2 = ||f|| 2 < ∞, it follows ̂f ∈ l 2 (Z). (ii) We claim that ||V f|| = ||f||: indeed, ||V f|| 2 = || ̂f|| 2 = ∑ | ̂f(n)| 2 = ||f|| 2 . If f = z n then { 0 if k ≠ n ̂f(k) = 1 if k = n, so that ̂f is the n-th basis vector in l 2 (Z). Thus ran V is dense and hence V is an isomorphism. (iii) If {e n } is an orthonormal basis for l 2 (Z) then by (ii), V z n = e n . Thus V W z n = V (z n+1 ) = e n+1 = UV z n . If T ∈ B(H), write Lat T for the set of all invariant subspaces for T , i.e., Lat T := {M ⊂ H : T M ⊂ M}. Theorem 5.1.2. If µ is a compactly supported measure on T and M ∈ LatN µ then M = ϕH 2 ⊕ L 2 (µ|∆), where ϕ ∈ L ∞ (µ) and ∆ is a Borel set of T such that ϕ|∆ = 0 a.e. and |ϕ| 2 µ = m(:=the normalized Lebesgue measure). Proof. See [Con3, p.121]. Now consider the case where µ = m (in this case, N µ is the bilateral shift). Observe ϕ ∈ L 2 , |ϕ| 2 m = m =⇒ |ϕ| = 1 a.e., so that there is no Borel set ∆ such that ϕ|∆ = 0 and m(∆) ≠ 0. Therefore every invariant subspace for the bilateral shift must have one form or the other. We thus have: Corollary 5.1.3. If W is the bilateral shift on L 2 and M ∈ Lat W then either M = L 2 (m|∆) or M = ϕH 2 for a Borel set ∆ and a function ϕ ∈ L ∞ such that |ϕ| = 1 a.e. Definition 5.1.4. A function ϕ ∈ L ∞ function if |ϕ| = 1 a.e. [ϕ ∈ H ∞ ] is called a unimodular [inner] The following theorem has had an enormous influence on the development in operator theory and function theory. Theorem 5.1.5 (Beurling’s Theorem). If U is the unilateral shift on H 2 then Lat U = {ϕH 2 : ϕ is an inner function}. 158
CHAPTER 5. TOEPLITZ THEORY Proof. Let W be the bilateral shift on L 2 . If M ∈ Lat U then M ∈ Lat W . By Corollary 5.1.3, M = L 2 (m|∆) or M = ϕH 2 , where ϕ is a unimodular function. Since U is a shift, ∩ U n M ⊂ ∩ U n H 2 = {0}, so the first alternative is impossible. Hence ϕH 2 = M ⊂ H 2 . Since ϕ = ϕ · 1 ∈ M, it follows ϕ ∈ L ∞ ∩ H 2 = H ∞ . 159
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1 Graduate Texts ——————
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3 . Preface The present lectures ar
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CONTENTS 4.4 The Perturbations . .
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CHAPTER 1. FREDHOLM THEORY 1.2 Prel
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CHAPTER 2. WEYL THEORY Problem 2.4.
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BIBLIOGRAPHY [AT] S.C. Arora and J.
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BIBLIOGRAPHY [Cu2] [Cu3] [CuD] [CuF
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BIBLIOGRAPHY [Fin] J.K. Finch, The
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BIBLIOGRAPHY [Ist] [IW] [KL] [JeL]
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BIBLIOGRAPHY [Smu] [Sn] J.L. Smul
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Woo Young Lee Lecture Notes on Operator Theory

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