Woo Young Lee Lecture Notes on Operator Theory

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CHAPTER 5. TOEPLITZ THEORY Theorem 5.2.6. ⎡ ⎤ c 0 c 1 c 0 O CSIP is solvable ⇐⇒ C ≡ c 2 c 1 c 0 ⎢ ⎣ . . . .. . ⎥ .. ⎦ c N−1 c N−2 · · · c 1 c 0 is a contraction. Moreover, φ is a solution if and only if T φ is a contractive lifting of C which commutes with the unilateral shift. Proof. (⇒) Assume that we have a solution φ. Then the condition (ii) implies ⎡ ⎤ φ 0 T φ = φ 1 φ 0 O ⎢ ⎣φ 2 φ 1 φ 0 ⎥ ⎦ . . . .. . .. (φ j := ̂φ(j)) is a contraction because ||T φ || = ||φ|| ∞ ≤ 1. So the compression of T φ is also contractive. In particular, ⎡ ⎤ φ 0 φ 1 φ 0 O ⎢ ⎣ . . ⎥ . .. ⎦ φ n−1 φ n−2 · · · φ 0 must have norm less than or equal to 1 for all n. Therefore if CSIP is solvable, then ||C|| ≤ 1. (⇐) Let ⎡ ⎤ c 0 c 1 c 0 O C ≡ c 2 c 1 c 0 with ||C|| ≤ 1 ⎢ ⎣ . . . .. . ⎥ .. ⎦ c N−1 c N−2 · · · c 1 c 0 and let ⎡ ⎤ 0 1 0 A := 1 0 : C N → C N . ⎢ ⎣ . .. . .. ⎥ ⎦ 1 0 Then A and C are contractions and AC = CA. Observe that the unilateral shift U is the minimal isometric dilation of A (please check it!). By the Commutant Lifting Theorem, C can be lifted to a contraction S such that SU = US. But then S is an 174
CHAPTER 5. TOEPLITZ THEORY analytic Toeplitz operator, i.e., S = T φ with φ ∈ H ∞ . Since S is a lifting of C we must have ̂φ(j) = c j (j = 0, 1, · · · , N − 1). Since S is a contraction, it follows that ||φ|| ∞ = ||T φ || ≤ 1. Now suppose φ is a trigonometric polynomial of the form φ(z) = N∑ n=−N a n z n (a N ≠ 0). If a function k ∈ H ∞ (T) satisfies φ − kφ ∈ H ∞ then k necessarily satisfies N∑ N∑ k a n z −n − a −n z −n ∈ H ∞ . (5.5) n=1 n=1 From (5.5) one compute the Fourier coefficients ̂k(0), · · · , ̂k(N − 1) to be ̂k(n) = c n (n = 0, 1, · · · , N − 1), where c 0 , c 1 , · · · , c N−1 are determined uniquely from the coefficients of φ by the following relation ⎡ c ⎡ ⎤−1 0 a 1 a 2 a 3 · · · a ⎡ N a −1 ⎢ c 1.. ⎥ ⎢ a 2 a 3 . . . · ⎥ ⎢ ⎢⎢⎢⎢⎢⎣ a −2 ⎥ ⎤ = ⎢ ⎥ ⎢ ⎣ . ⎦ ⎣ c N−1 a 3 . . . · · · . · · · O a N Thus if k(z) = ∑ ∞ j=0 c jz j is a function in H ∞ then φ − kφ ∈ H ∞ ⇐⇒ c 0 , c 1 , · · · , c N−1 are given by (5.6). ⎥ ⎦ ⎤ . . (5.6) ⎥ . ⎦ a −N Thus by Cowen’s theorem, if c 0 , c 1 , · · · , c N−1 are given by (5.6) then the hyponormality of T φ is equivalent to the existence of a function k ∈ H ∞ such that {̂k(j) = cj (j = 0, · · · , N − 1) ||k|| ∞ ≤ 1, which is precisely the formulation of CSIP. Therefore we have: Theorem 5.2.7. If φ(z) = ∑ N n=−N a nz n , where a N ≠ 0 and if c 0 , c 1 , · · · , c N−1 are given by (5.6) then ⎡ ⎤ c 0 c 1 c 0 O T φ is hyponormal ⇐⇒ C ≡ c 2 c 1 c 0 is a contraction. ⎢ ⎣ . . . .. . ⎥ .. ⎦ c N−1 c N−2 · · · c 1 c 0 175
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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 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 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. Newbu
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CHAPTER 2. WEYL THEORY Proof. By Le
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CHAPTER 2. WEYL THEORY Proof. If λ
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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 A question a
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CHAPTER 2. WEYL THEORY (see [Ta2, T
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CHAPTER 2. WEYL THEORY Problem 2.4.
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CHAPTER 4. WEIGHTED SHIFTS Since β
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CHAPTER 4. WEIGHTED SHIFTS then c(n
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where ⎧⎪⎨ ⎪ ⎩ q n = (α 2
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CHAPTER 4. WEIGHTED SHIFTS 4.4 The
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Woo Young Lee Lecture Notes on Operator Theory

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