Woo Young Lee Lecture Notes on Operator Theory

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CHAPTER 4. WEIGHTED SHIFTS 156
Chapter 5 Toeplitz Theory 5.1 Preliminaries 5.1.1 Fourier Transform and Beurling’s Theorem A trigonometric polynomial is a function p ∈ C(T) of the form ∑ n k=−n a kz k . It was well-known that the set of trigonometric polynomials is uniformly dense in C(T) and hence is dense in L 2 (T). In fact, if e n := z n , (n ∈ Z) then {e n : n ∈ Z} forms an orthonormal basis for L 2 (T). The Hardy space H 2 (T) is spanned by {e n : n = 0, 1, 2, · · · }. Write H ∞ (T) := L ∞ (T) ∩ H 2 (T). Then H ∞ is a subalgebra of L ∞ . Let m :=the normalized Lebesgue measure on T and write L 2 := L 2 (T). If f ∈ L 2 then the Fourier transform of f, ̂f : Z → C, is defined by ∫ ̂f(n) ≡ ⟨f, e n ⟩ = T fz n dm = 1 2π ∫ 2π 0 f(t)e −int dt, which is called the n-th Fourier coefficient of f. By Parseval’s identity, ∞∑ f = ̂f(n)z n , n=−∞ which converges in the norm of L 2 . This series is called the Fourier series of f. Proposition 5.1.1. We have: (i) f ∈ L 2 ⇒ ̂f ∈ l 2 (Z); (ii) If V : L 2 → l 2 (Z) is defined by V f = ̂f then V is an isomorphism. (iii) If W = N m on L 2 then V W V −1 is the bilateral shift on l 2 (Z). 157
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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 1. FREDHOLM THEORY 1.3 Defi
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CHAPTER 1. FREDHOLM THEORY Corollar
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CHAPTER 1. FREDHOLM THEORY 1.6 Pert
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CHAPTER 1. FREDHOLM THEORY The Weyl
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CHAPTER 1. FREDHOLM THEORY Theorem
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CHAPTER 1. FREDHOLM THEORY Since T
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CHAPTER 1. FREDHOLM THEORY Proof. (
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CHAPTER 1. FREDHOLM THEORY The foll
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CHAPTER 1. FREDHOLM THEORY 1.10 Ess
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CHAPTER 1. FREDHOLM THEORY In fact,
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CHAPTER 1. FREDHOLM THEORY Since T
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CHAPTER 1. FREDHOLM THEORY 1.13 Com
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CHAPTER 2. WEYL THEORY finite dimen
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CHAPTER 2. WEYL THEORY We shall cal
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CHAPTER 2. WEYL THEORY If the condi
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CHAPTER 2. WEYL THEORY are called q
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CHAPTER 2. WEYL THEORY Proof. Let
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CHAPTER 2. WEYL THEORY which tends
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CHAPTER 2. WEYL THEORY We thus have
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CHAPTER 2. WEYL THEORY Proof. If λ
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CHAPTER 2. WEYL THEORY Evidently K
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CHAPTER 2. WEYL THEORY Example 2.4.
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CHAPTER 2. WEYL THEORY (ii) σ(T )
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CHAPTER 2. WEYL THEORY 2.5 Weyl’s
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CHAPTER 2. WEYL THEORY (see [Ta2, T
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CHAPTER 2. WEYL THEORY Corollary 2.
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CHAPTER 2. WEYL THEORY Problem 2.4.
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CHAPTER 6. A BRIEF SURVEY ON THE IN
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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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