Woo Young Lee Lecture Notes on Operator Theory

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CHAPTER 3. HYPONORMAL AND SUBNORMAL THEORY Since {P k } is an increasing sequence, tr[T ∗ , T ] = lim k tr ( P k [T ∗ ) , T ]P k . By Lemma 3.2.4 we get ) tr [T ∗ , T ] ≤ limsup∥Pk ⊥ T P k ∥ 2 2 ≤ limsup (m∥Pk ⊥ T P k ∥ 2 ≤ m∥T ∥ 2 . We are ready for: Proof of the Berger-Shaw Theorem. Let R = ∥T ∥ and put D = B(0; R). If ε > 0, let D 1 , · · · , D n be pairwise disjoint closed disks contained in D \ σ(T ) such that Area (D) < Area σ(T ) + ∑ j Area (D j ) + ε. If D j = B(a j ; r j ), this inequality says πR 2 − π ∑ j r 2 j < Area σ(T ) + ε. If S is the unilateral shift of multiplicity 1, let S j = (a j + r j S) (m) . Now that each S j is m-multicyclic. Thus ⎡ ⎤ T S 1 0 A = ⎢ ⎣ . 0 .. ⎥ ⎦ S n is an m-multicyclic hyponormal operator since the spectra of the operator summands are pairwise disjoint. Also ∥A∥ = R. By Lemma 3.2.6, tr [A ∗ , A] ≤ mR 2 . But Therefore tr [A ∗ , A] = tr [T ∗ , T ] + ⎛ n∑ n∑ tr [Sj ∗ , S j ] = tr [T ∗ , T ] + m rj 2 . j=1 π tr [T ∗ , T ] ≤ m ⎝πR 2 − π n∑ j=1 Since ε was arbitrary, the proof is complete. r 2 j ⎞ j=1 ⎠ ≤ m ( Area σ(T ) + ε ) . Theorem 3.2.7. (Putnam’s inequality) If S ∈ B(H) is a hyponormal operator then ∥[S ∗ , S]∥ ≤ 1 Area (σ(S)). π 84
CHAPTER 3. HYPONORMAL AND SUBNORMAL THEORY Proof. Fix ∥f∥ ≤ 1 and let K ≡ ∨ {r(s)f : r ∈ Rat (σ(S))}. If T = S| K then T is an 1-multicyclic hyponormal operator. By the Berger-Shaw theorem and the fact that ∥T ∗ f∥ ≤ ∥S ∗ f∥, we get Since f was arbitrary, the result follows. ⟨[S ∗ , S]f, f⟩ = ∥Sf∥ 2 − ∥S ∗ f∥ 2 ≤ ∥T f∥ 2 − ∥T ∗ f∥ 2 = ⟨[T ∗ , T ]f, f⟩ ≤ tr [T ∗ , T ] ≤ 1 Area(σ(T )) π ≤ 1 π Area(σ(S)). Corollary 3.2.8. If S is a hyponormal operator such that Area (σ(S)) = 0 then S is normal. 85
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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 The Weyl
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CHAPTER 1. FREDHOLM THEORY Theorem
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CHAPTER 1. FREDHOLM THEORY The foll
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CHAPTER 5. TOEPLITZ THEORY Proof. (
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CHAPTER 5. TOEPLITZ THEORY From (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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