Woo Young Lee Lecture Notes on Operator Theory

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CHAPTER 2. WEYL THEORY finite dimensional component and a component modulo finite dimensions. Weyl’s theorem asks not just that the spectrum split into Fredholm spectra and eigenvalues: it wants the spectrum to divide into Weyl spectrum and eigenvalues which are both topologically isolated in the spectrum, and geometrically of finite multiplicity, with finite dimensional eigenspaces. 40
CHAPTER 2. WEYL THEORY 2.2 Weyl’s Theorem If T ∈ B(X) write π 0f (T ) for the eigenvalues of finite multiplicity; π 0i (T ) for the eigenvalues of infinite multiplicity; N(T ) and R(T ) for the null space and the range of T , respectively. If we write iso K = K \ acc K, and ∂ K for the topological boundary of K, and π 00 (T ) := {λ ∈ iso σ(T ) : 0 < dim N(T − λI) < ∞} (2.1) for the isolated eigenvalues of finite multiplicity, and ([Har4]) p 00 (T ) := σ(T ) \ σ b (T ) (2.2) for the Riesz points of σ(T ), then by the punctured neighborhood theorem, i.e., ∂ σ(T ) \ σ e (T ) ⊆ iso σ(T ) (cf. [Har4], [HaL1]), iso σ(T ) \ σ e (T ) = iso σ(T ) \ ω(T ) = p 00 (T ) ⊆ π 00 (T ). (2.3) H. Weyl [We] examined the spectra of all compact perturbations T + K of a single hermitian operator T and discovered that λ ∈ σ(T + K) for every compact operator K if and only if λ is not an isolated eigenvalue of finite multiplicity in σ(T ). Today this result is known as Weyl’s theorem: that is, we say that Weyl’s theorem holds for T ∈ B(X) if there is equality σ(T ) \ ω(T ) = π 00 (T ). (2.4) In this section we explore the class of operators satisfying Weyl’s theorem. If T ∈ B(X), write r(T ) for the spectral radius of T . It is familiar that r(T ) ≤ ||T ||. An operator T is called normaloid if r(T ) = ||T || and isoloid if iso σ(T ) ⊆ π 0 (T ). If X is a Hilbert space, an operator T ∈ B(X) is called reduction-isoloid if the restriction of T to any reducing subspace is isoloid. Let X be a Hilbert space and suppose that T ∈ B(X) is reduced by each of its finite-dimensional eigenspaces. If M := ∨ {N(T − λI) : λ ∈ π 0f (T )}, then M reduces T . Let T 1 := T |M and T 2 := T |M ⊥ . Then we have ([Be2, Proposition 4.1]) that (i) T 1 is a normal operator with pure point spectrum; (ii) π 0 (T 1 ) = π 0f (T ); (iii) σ(T 1 ) = cl π 0 (T 1 ); (iv) π 0 (T 2 ) = π 0 (T ) \ π 0f (T ) = π 0i (T ). In this case, S.Berberian ([Be2, Definition 5.4]) defined τ(T ) := σ(T 2 ) ∪ acc π 0f (T ). (2.5) 41
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CHAPTER 4. WEIGHTED SHIFTS Observe
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CHAPTER 4. WEIGHTED SHIFTS Since β
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where ⎧⎪⎨ ⎪ ⎩ q n = (α 2
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CHAPTER 4. WEIGHTED SHIFTS One migh
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CHAPTER 5. TOEPLITZ THEORY Proof. (
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CHAPTER 5. TOEPLITZ THEORY Remark 5
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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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