Woo Young Lee Lecture Notes on Operator Theory

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CHAPTER 3. HYPONORMAL AND SUBNORMAL THEORY Then Q is a POM and is supported on σ(N). Also for all h ∈ H, ⟨S ∗n S m h, h⟩ = ⟨N ∗n N m h, h⟩ ∫ = z n z m d⟨E(z)h, h⟩ ∫ = z n z m d⟨E(z)h, P h⟩ ∫ = z n z m d⟨P E(z)h, h⟩ ∫ = z n z m d⟨Q(z)h, h⟩ ⟨(∫ ) ⟩ = z n z m dQ(z) h, h , so that ∫ S ∗n S m = z n z m dQ(z). (e) ⇒ (f): Let Q be the POM hypothesized in (i) and K = supp Q. For a Borel set △ ⊆ [0, ∞), define Q + (△) := Q{z ∈ C : |z| ∈ △}. In fact, Q + (△) = Q(τ −1 △), where τ(z) = |z|. Then Q + is a POM whose support ⊆ [0, a] with a = max z∈K |z|. For any f ∈ H, ∫ ∫ t 2n d⟨Q + (t)f, f⟩ = |z| 2n d⟨Q(z)f, f⟩. (f) ⇒ (c): Fix f 0 , · · · , f n ∈ H and define scalar-valued measures µ jk by µ jk (△) = ⟨Q(△)f j , f k ⟩. Let µ be a positive measure on [0, a] such that µ jk ≪ µ for any j, k. Let h jk = dµ jk dµ (Radon-Nikodym derivative). For each u ∈ C[0, a], ρ(u) = ∫ udQ defines a bounded operator and ρ : C[0, a] → B(H) is a positive linear map. Note that for all u, ⟨ρ(u)f j , f k ⟩ = ∫ udµ jk = ∫ uh j,k dµ. Moreover if λ 0 , · · · , λ n ∈ C and u ≥ 0 then ∑ (∫ j,k ) uh jk dµ λ j λ k = ∥ ∑ ∥∥∥∥∥ 2 ρ(u) 1 2 λj f j ≥ 0. ∥ It follows that (h jk (t)) j,k is positive (n + 1) × (n + 1) matrix for [µ] almost every t. This implies that ∑ h jk (t)t 2j t 2k ≥ 0 a.e. [µ]. j,k 90 j
= ∑ j,k CHAPTER 3. HYPONORMAL AND SUBNORMAL THEORY Therefore ∫ ∑ 0 ≤ j,k = ∑ ∫ j,k h jk (t)t 2(j+k) dµ(t) t 2(j+k) dµ jk (t) ⟨ S j+k f j , S j+k f k ⟩ , so (3.4) holds. Remark. Without loss of generality we may assume that ||S|| < 1. Let K = H ∞ and let K 0 = the finitely nonzero sequences in K. Let ⎡ 1 S ∗ S ∗2 ⎤ · · · S S ∗ S S ∗2 S · · · M = S 2 S ∗ S 2 S ∗2 S 2 · · · ⎢ ⎣S 3 S ∗ S 3 S ∗2 S 3 · · · ⎥ ⎦ . . . on K 0 . If f = (f 0 , · · · , f n , · · · ) ∈ K 0 then ∑ ∥(Mf) j ∥ 2 = ∑ j j ≤ ∑ j ∥ ∑ ∥∥∥∥ 2 S ∗k S j f ∥ k k [ ] ∑ ∥S∥ k+j ∥f k ∥ 2 k ≤ ∑ j [ ∑ k ∥S∥ 2k+2j ] [ ∑ k ∥f k ∥ ] 2 ≤ (1 − ∥S∥ 2 ) −2 ∥f∥ 2 . Since ∥S∥ < 1, Mf ∈ K and M extends to a bounded operator on K. Clearly, M is hermitian. Note ⟨Mf, f⟩ K = ∑ ⟨S j f k , S k f j ⟩. j,k So (3.3) holds ⇐⇒ M is positive. [ ] A B Recall the Smul’jan theorem – if M = B ∗ (A, C hermitian, A invertible), then C M ≥ 0 ⇐⇒ B ∗ A −1 B ≤ C. 91
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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 1.5 The
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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 4. WEIGHTED SHIFTS In fact,
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CHAPTER 4. WEIGHTED SHIFTS H k+1 ,
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CHAPTER 4. WEIGHTED SHIFTS are both
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CHAPTER 4. WEIGHTED SHIFTS Write f
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CHAPTER 4. WEIGHTED SHIFTS Proof. A
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CHAPTER 4. WEIGHTED SHIFTS We remem
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CHAPTER 5. TOEPLITZ THEORY Proof. (
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CHAPTER 5. TOEPLITZ THEORY 5.1.2 Ha
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CHAPTER 5. TOEPLITZ THEORY Proof. I
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CHAPTER 5. TOEPLITZ THEORY where ϕ
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CHAPTER 5. TOEPLITZ THEORY Proof. I
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CHAPTER 5. TOEPLITZ THEORY From (5.
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CHAPTER 5. TOEPLITZ THEORY Remark 5
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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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