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r r Prtrts r - - PowerPoint PPT Presentation

Oct 27, 2023 399 likes •644 views

r r Prtrts r t rtrs rst r ts

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SLIDE 1

❙✉♣❡r ❙✐♥❣✉❧❛r P❡rt✉r❜❛t✐♦♥s ❢♦r ◆♦♥✲❙❡♠✐❜♦✉♥❞❡❞ ❙❡❧❢✲❆❞❥♦✐♥t ❖♣❡r❛t♦rs

❈❤r✐st♦♣❤ ◆❡✉♥❡r ❙t♦❝❦❤♦❧♠s ❯♥✐✈❡rs✐t❡t✱ ❙✇❡❞❡♥ ❲❡✐❤♥❛❝❤ts❦♦❧❧♦q✉✐✉♠ ✭❖❚■◆❉✮ ❚❯ ❱✐❡♥♥❛ ✷✵✶✻

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SLIDE 2

❖✈❡r✈✐❡✇

  • ❡♥❡r❛❧✿ ●♦❛❧ ✫ ❑♥♦✇♥ ❘❡s✉❧ts

❋❡❛t✉r❡ ✭■✮✿ ❍✐❧❜❡rt s♣❛❝❡ ❋❡❛t✉r❡✭■■✮✿ ❑r❡✐♥✬s ❢♦r♠✉❧❛ ❏♦✐♥t ✇♦r❦ ✇✐t❤ ❆♥♥❡♠❛r✐❡ ▲✉❣❡r ❛♥❞ P❛✈❡❧ ❑✉r❛s♦✈

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SLIDE 3
  • ♦❛❧

▲❡t

◮ A = A∗ ✐♥ ❛ ❍✐❧❜❡rt s♣❛❝❡ H✱ ♥♦♥✲s❡♠✐❜♦✉♥❞❡❞

❀ ✐♥t❡r❡st✐♥❣ ❝❛s❡ σ(A) = R

◮ ❧❡t ϕ ∈ H−n(A) ❢♦r n ≥ ✸

✭✧s✉♣❡r s✐♥❣✉❧❛r✧✮ ❆✐♠✿ ●✐✈❡ ❛♥ ✐♥t❡r♣r❡t❛t✐♦♥ ♦❢ t❤❡ ❢♦r♠❛❧ s✉♠ Aα := A + αϕ, ·ϕ α ∈ R ∪ {∞} ✐♥ ❛ ❍✐❧❜❡rt s♣❛❝❡ ♠♦❞❡❧

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SLIDE 4

❙♦♠❡ t❡❝❤♥✐❝❛❧✐t✐❡s✿ ❙❝❛❧❡ ♦❢ ❍✐❧❜❡rt ❙♣❛❝❡s

❉❡✜♥❡

◮ H✵(A) := H ✇✐t❤ ♥♦r♠ · ✵ ◮ H✷(A) := ❞♦♠ A ❡q✉✐♣♣❡❞ ✇✐t❤ t❤❡ ❣r❛♣❤ ♥♦r♠ · ✷ := (|A| + ✶) · ✵

❀ (H✷(A), · ✷) ✐s ❝♦♠♣❧❡t❡

◮ s✐♠✐❧❛r❧②✿ Hn(A) ✇✐t❤ ♥♦r♠ · n := (|A| + ✶)n/✷ · ✵ ❢♦r n ≥ ✵ ◮ H−n(A) := Hn(A)∗ ❢♦r ♥❡❣❛t✐✈❡ ✐♥❞✐❝❡s✱ ✇✐t❤ s✐♠✐❧❛r ♥♦r♠

· · · ⊇ H−n ⊇ · · · ⊇ H−✷ ⊇ H−✶ ⊇ H✵ ⊇ H✶ ⊇ H✷ ⊇ · · · ⊇ Hn ⊇ · · · ❖r✐❣✐♥❛❧❧②✱ (A − z) : H✷ → H✵ ❛♥❞ (A − z)−✶ : H✵ → H✷ ❀ ❝❛♥ ❜❡ ❡①t❡♥❞❡❞ t♦ t❤❡ s❝❛❧❡ s✳t✳✿ (A − z) : Hn → Hn−✷ ❛♥❞ (A − z)−✶ : Hn → Hn+✷ ❢♦r n ∈ Z

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SLIDE 5

❙♦♠❡ t❡❝❤♥✐❝❛❧✐t✐❡s✿ ❙❝❛❧❡ ♦❢ ❍✐❧❜❡rt ❙♣❛❝❡s

❉❡✜♥❡

◮ H✵(A) := H ✇✐t❤ ♥♦r♠ · ✵ ◮ H✷(A) := ❞♦♠ A ❡q✉✐♣♣❡❞ ✇✐t❤ t❤❡ ❣r❛♣❤ ♥♦r♠ · ✷ := (|A| + ✶) · ✵

❀ (H✷(A), · ✷) ✐s ❝♦♠♣❧❡t❡

◮ s✐♠✐❧❛r❧②✿ Hn(A) ✇✐t❤ ♥♦r♠ · n := (|A| + ✶)n/✷ · ✵ ❢♦r n ≥ ✵ ◮ H−n(A) := Hn(A)∗ ❢♦r ♥❡❣❛t✐✈❡ ✐♥❞✐❝❡s✱ ✇✐t❤ s✐♠✐❧❛r ♥♦r♠

· · · ⊇ H−n ⊇ · · · ⊇ H−✷ ⊇ H−✶ ⊇ H✵ ⊇ H✶ ⊇ H✷ ⊇ · · · ⊇ Hn ⊇ · · · ❖r✐❣✐♥❛❧❧②✱ (A − z) : H✷ → H✵ ❛♥❞ (A − z)−✶ : H✵ → H✷ ❀ ❝❛♥ ❜❡ ❡①t❡♥❞❡❞ t♦ t❤❡ s❝❛❧❡ s✳t✳✿ (A − z) : Hn → Hn−✷ ❛♥❞ (A − z)−✶ : Hn → Hn+✷ ❢♦r n ∈ Z

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SLIDE 6

❙♦♠❡ t❡❝❤♥✐❝❛❧✐t✐❡s✿ ❙❝❛❧❡ ♦❢ ❍✐❧❜❡rt ❙♣❛❝❡s

❉❡✜♥❡

◮ H✵(A) := H ✇✐t❤ ♥♦r♠ · ✵ ◮ H✷(A) := ❞♦♠ A ❡q✉✐♣♣❡❞ ✇✐t❤ t❤❡ ❣r❛♣❤ ♥♦r♠ · ✷ := (|A| + ✶) · ✵

❀ (H✷(A), · ✷) ✐s ❝♦♠♣❧❡t❡

◮ s✐♠✐❧❛r❧②✿ Hn(A) ✇✐t❤ ♥♦r♠ · n := (|A| + ✶)n/✷ · ✵ ❢♦r n ≥ ✵ ◮ H−n(A) := Hn(A)∗ ❢♦r ♥❡❣❛t✐✈❡ ✐♥❞✐❝❡s✱ ✇✐t❤ s✐♠✐❧❛r ♥♦r♠

· · · ⊇ H−n ⊇ · · · ⊇ H−✷ ⊇ H−✶ ⊇ H✵ ⊇ H✶ ⊇ H✷ ⊇ · · · ⊇ Hn ⊇ · · · ❖r✐❣✐♥❛❧❧②✱ (A − z) : H✷ → H✵ ❛♥❞ (A − z)−✶ : H✵ → H✷ ❀ ❝❛♥ ❜❡ ❡①t❡♥❞❡❞ t♦ t❤❡ s❝❛❧❡ s✳t✳✿ (A − z) : Hn → Hn−✷ ❛♥❞ (A − z)−✶ : Hn → Hn+✷ ❢♦r n ∈ Z

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SLIDE 7

❙♦♠❡ ❑♥♦✇♥ ❘❡s✉❧ts ❢♦r Aα = A + αϕ, ·ϕ

◮ ϕ ∈ H✱ A ❜♦✉♥❞❡❞ ❀ ❞❡s❝r✐❜❡ r❡s♦❧✈❡♥ts ✐♥ r❡❢❡r❡♥❝❡ t♦ A = A✵

(Aα − z)−✶ = (A − z)−✶ − (A − z)−✶ϕ, · q(z) − ✶

α

(A − z)−✶ϕ q(z) = ϕ, (A − z)−✶ϕ ✭◆❡✈❛♥❧✐♥♥❛ ❢✉♥❝t✐♦♥✮ str❛✐❣❤❢♦r✇❛r❞ ❝❛❧❝✉❧❛t✐♦♥✦ ✧✐♥✜♥✐t❡ ❝♦✉♣❧✐♥❣✧✿ α → ∞ ✭❧✐♥❡❛r r❡❧❛t✐♦♥✮

◮ ϕ ❛t ♠♦st ❢r♦♠ H−✷ ✭✧s✐♥❣✉❧❛r✧✮

❀ ❞❡✜♥❡ ❛ s②♠♠❡tr② ✧S := A|ϕ⊥✧✱ ❝❧♦s❡❞✱ ❞❡♥s❡❧② ❞❡✜♥❡❞✱ ❞❡❢❡❝t (✶, ✶) ❀ ❞❡❢❡❝t ❡❧❡♠❡♥t ϕz✱ Q✲❢✉♥❝t✐♦♥ Q(z) ❢r♦♠ Q(z)−Q(z✵)

z−z✵

= ϕz✵, ϕz ❛❧❧ s✳❛✳ ❡①t❡♥s✐♦♥s ♦❢ S ❛r❡ ♣❛r❛♠❡tr✐③❡❞ ❜② ❑r❡✐♥✬s ❢♦r♠✉❧❛ (Aτ − z)−✶ = (A − z)−✶ − (A − z)−✶ϕ, · Q(z) + ✶

τ

(A − z)−✶ϕ ❝♦♥♥❡❝t✐♦♥ ❜❡t✇❡❡♥ τ ❛♥❞ α✿ ❡①♣❧✐❝✐t ♦♥❧② ❢♦r ϕ ∈ H−✶✱ ♦t❤❡r✇✐s❡ (Aα)α∈R∪{∞} = (Aτ)τ∈R∪{∞} ❛s ❢❛♠✐❧✐❡s

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SLIDE 8

❙♦♠❡ ❑♥♦✇♥ ❘❡s✉❧ts ❢♦r Aα = A + αϕ, ·ϕ

◮ ϕ ∈ H✱ A ❜♦✉♥❞❡❞ ❀ ❞❡s❝r✐❜❡ r❡s♦❧✈❡♥ts ✐♥ r❡❢❡r❡♥❝❡ t♦ A = A✵

(Aα − z)−✶ = (A − z)−✶ − (A − z)−✶ϕ, · q(z) − ✶

α

(A − z)−✶ϕ q(z) = ϕ, (A − z)−✶ϕ ✭◆❡✈❛♥❧✐♥♥❛ ❢✉♥❝t✐♦♥✮ str❛✐❣❤❢♦r✇❛r❞ ❝❛❧❝✉❧❛t✐♦♥✦ ✧✐♥✜♥✐t❡ ❝♦✉♣❧✐♥❣✧✿ α → ∞ ✭❧✐♥❡❛r r❡❧❛t✐♦♥✮

◮ ϕ ❛t ♠♦st ❢r♦♠ H−✷ ✭✧s✐♥❣✉❧❛r✧✮

❀ ❞❡✜♥❡ ❛ s②♠♠❡tr② ✧S := A|ϕ⊥✧✱ ❝❧♦s❡❞✱ ❞❡♥s❡❧② ❞❡✜♥❡❞✱ ❞❡❢❡❝t (✶, ✶) ❀ ❞❡❢❡❝t ❡❧❡♠❡♥t ϕz✱ Q✲❢✉♥❝t✐♦♥ Q(z) ❢r♦♠ Q(z)−Q(z✵)

z−z✵

= ϕz✵, ϕz ❛❧❧ s✳❛✳ ❡①t❡♥s✐♦♥s ♦❢ S ❛r❡ ♣❛r❛♠❡tr✐③❡❞ ❜② ❑r❡✐♥✬s ❢♦r♠✉❧❛ (Aτ − z)−✶ = (A − z)−✶ − (A − z)−✶ϕ, · Q(z) + ✶

τ

(A − z)−✶ϕ ❝♦♥♥❡❝t✐♦♥ ❜❡t✇❡❡♥ τ ❛♥❞ α✿ ❡①♣❧✐❝✐t ♦♥❧② ❢♦r ϕ ∈ H−✶✱ ♦t❤❡r✇✐s❡ (Aα)α∈R∪{∞} = (Aτ)τ∈R∪{∞} ❛s ❢❛♠✐❧✐❡s

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SLIDE 9

❙♦♠❡ ❑♥♦✇♥ ❘❡s✉❧ts ■■

ϕ ∈ H−n ❢♦r n ≥ ✸ ✭✧s✉♣❡r s✐♥❣✉❧❛r✧✮ ❛♥❞ A s❡♠✐❜♦✉♥❞❡❞✱ ✐✳❡✳✱ ✐♥❢ σ(A) > −∞ s❡✈❡r❛❧ ♠♦❞❡❧s t♦ ❞❡s❝r✐❜❡ t❤❡s❡ ♣❡rt✉r❜❛t✐♦♥s✱ ❡✳❣✳✿

◮ ❉✐❥❦s♠❛✱ ❙❤♦♥❞✐♥ ❀ ✐♥ P♦♥tr②❛❣✐♥ s♣❛❝❡ ◮ ❑✉r❛s♦✈ ❀ ❍✐❧❜❡rt s♣❛❝❡✦

■♥❣r❡❞✐❡♥ts ✐♥ t❤❡ ❍✐❧❜❡rt s♣❛❝❡ ♠♦❞❡❧✿

◮ µ✶, . . . , µn−✷, µ ∈ R✱ ❞✐st✐♥❝t✱ t♦ t❤❡ ❧❡❢t ♦❢ t❤❡ σ(A) ◮ ♣♦❧②♥♦♠✐❛❧ b(z) = n−✷ i=✶ (z − µi)

❀ b(z) > ✵ ♦♥ σ(A)

◮ ❞✐❛❣♦♥❛❧ ♠❛tr✐① M = ❞✐❛❣(µ✶, . . . , µn−✷) ◮ ❝♦❡✣❝✐❡♥ts ai ❢r♦♠ t❤❡ ♣❛rt✐❛❧ ❢r❛❝t✐♦♥ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ b(z)−✶ ◮ ❡❧❡♠❡♥ts g(µi) := (A − µi)−✶ϕ

∈ H−n+✷

◮ ♦♥❡ ✈❡r② r❡❣✉❧❛r ❡❧❡♠❡♥t G(µ) := (A − µ)−✶ n−✷ i=✶ (A − µi)−✶ϕ

∈ Hn−✷

◮ ❛ ♣♦s✐t✐✈❡ ❞❡✜♥✐t❡ s✳❛✳ ♠❛tr✐① Γ s✉❝❤ t❤❛t ΓM = MΓ

slide-10
SLIDE 10

❙♦♠❡ ❑♥♦✇♥ ❘❡s✉❧ts ■■

ϕ ∈ H−n ❢♦r n ≥ ✸ ✭✧s✉♣❡r s✐♥❣✉❧❛r✧✮ ❛♥❞ A s❡♠✐❜♦✉♥❞❡❞✱ ✐✳❡✳✱ ✐♥❢ σ(A) > −∞ s❡✈❡r❛❧ ♠♦❞❡❧s t♦ ❞❡s❝r✐❜❡ t❤❡s❡ ♣❡rt✉r❜❛t✐♦♥s✱ ❡✳❣✳✿

◮ ❉✐❥❦s♠❛✱ ❙❤♦♥❞✐♥ ❀ ✐♥ P♦♥tr②❛❣✐♥ s♣❛❝❡ ◮ ❑✉r❛s♦✈ ❀ ❍✐❧❜❡rt s♣❛❝❡✦

■♥❣r❡❞✐❡♥ts ✐♥ t❤❡ ❍✐❧❜❡rt s♣❛❝❡ ♠♦❞❡❧✿

◮ µ✶, . . . , µn−✷, µ ∈ R✱ ❞✐st✐♥❝t✱ t♦ t❤❡ ❧❡❢t ♦❢ t❤❡ σ(A) ◮ ♣♦❧②♥♦♠✐❛❧ b(z) = n−✷ i=✶ (z − µi)

❀ b(z) > ✵ ♦♥ σ(A)

◮ ❞✐❛❣♦♥❛❧ ♠❛tr✐① M = ❞✐❛❣(µ✶, . . . , µn−✷) ◮ ❝♦❡✣❝✐❡♥ts ai ❢r♦♠ t❤❡ ♣❛rt✐❛❧ ❢r❛❝t✐♦♥ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ b(z)−✶ ◮ ❡❧❡♠❡♥ts g(µi) := (A − µi)−✶ϕ

∈ H−n+✷

◮ ♦♥❡ ✈❡r② r❡❣✉❧❛r ❡❧❡♠❡♥t G(µ) := (A − µ)−✶ n−✷ i=✶ (A − µi)−✶ϕ

∈ Hn−✷

◮ ❛ ♣♦s✐t✐✈❡ ❞❡✜♥✐t❡ s✳❛✳ ♠❛tr✐① Γ s✉❝❤ t❤❛t ΓM = MΓ

slide-11
SLIDE 11

❍✐❧❜❡rt s♣❛❝❡ ♠♦❞❡❧ ✭❑✉r❛s♦✈✮

■❞❡❛✿ ♠❛❦❡ H s♠❛❧❧❡r s✉❝❤ t❤❛t ϕ, · ♠❛❦❡s s❡♥s❡✱ ❜✉t ❛❞❞ ❡❧❡♠❡♥ts ♦❢ t❤❡ ❢♦r♠ (A − z)−✶ϕ ❉❡✜♥❡ H := Hn−✷ ⊕ Cn−✷ U, V := U, b(A)V + u, Γ v ◆♦t❡✿ H ֒ → H−n+✷ A ❣✐✈❡s r✐s❡ t♦ t✇♦ ♦♣❡r❛t♦rs✿ ✶✳ ❞♦♠ A♠❛① := {U ∈ H : U = Ur + uG(µ), Ur ∈ Hn, u ∈ C, u ∈ Cn−✷} A♠❛①  Ur + uG(µ)

  • u

  =  AUr + µuG(µ) M u + u a   ✷✳ ❞♦♠ A♠✐♥ := {U ∈ ❞♦♠ A♠❛① : u = ✵, Γ a, u = ϕ, Ur} A♠✐♥  Ur

  • u

  =   AUr Γ−✶MΓ u   =  AUr M u  

slide-12
SLIDE 12

❍✐❧❜❡rt ❙♣❛❝❡ ▼♦❞❡❧ ■■

❇② ❞❡✜♥✐t✐♦♥✱ A♠✐♥ = A∗

♠❛①✱ ✐t ✐s s②♠♠❡tr✐❝ ✇✐t❤ ❞❡❢❡❝t (✶, ✶)✱ ❞❡❢❡❝t ❡❧❡♠❡♥t

Φ(z) =   G(µ) −(M − z)−✶ a   ❀ ❡♠❜❡❞❞❡❞✿ ✶ b(z)(A − z)−✶ϕ ❀ ❑r❡✐♥✬s ❢♦r♠✉❧❛ ✇✐t❤ A✵ := A ⊕ M ❛s s✳❛✳ r❡❢❡r❡♥❝❡ ♦♣❡r❛t♦r ✰ r❡str✐❝t✐♦♥✴❡♠❜❡❞❞✐♥❣ ❣✐✈❡s ι(Aθ − z)−✶|Hn−✷ = (A − z)−✶ − (A − z)−✶ϕ, · Q(z) + ❝♦t θ ✶ b(z)(A − z)−✶ϕ Q✲❢✉♥❝t✐♦♥✿ Q(z) = ϕ, (z − µ)(A − µ)−✶G(z) + a, Γ(M − z)−✶ a ❇♦♥✉s✿ ❞❡s❝r✐♣t✐♦♥ ♦❢ t❤❡ ❢❛♠✐❧② (Aθ)θ∈[✵,π) ✈✐❛ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s

slide-13
SLIDE 13

❈♦♥❝❧✉s✐♦♥

❚❤❡ ❞❡s❝r✐❜❡❞ ♠♦❞❡❧ ❤❛s t✇♦ ❦❡② ❢❡❛t✉r❡s✿ ✭■✮ (H, ·, ·H) ✐s ❛ ❍✐❧❜❡rt s♣❛❝❡ ✭■■✮ ❑r❡✐♥ t②♣❡ ❢♦r♠✉❧❛ t♦ ❞❡s❝r✐❜❡ ❛❧❧ ♣❡rt✉r❜❛t✐♦♥s ✐♥ H ◗✉❡st✐♦♥✿ ❝❛♥ ✇❡ ❞❡s❝r✐❜❡ t❤❡ ♣❡rt✉r❜❛t✐♦♥s A + αϕ, ·ϕ ✇❤❡♥ A ✐s ♥♦ ❧♦♥❣❡r s❡♠✐❜♦✉♥❞❡❞ ❛♥❞ ❦❡❡♣ ❜♦t❤ ❢❡❛t✉r❡s (I) ❛♥❞ (II)❄ ❘❡♠❛r❦✿ ■❢ σ(A) ❤❛s ❣❛♣s✱ t❤❡ ♣r❡✈✐♦✉s ♠♦❞❡❧ st✐❧❧ ❛♣♣❧✐❡s✳ ❚❤❡ ✐♥t❡r❡st✐♥❣ ❝❛s❡ ✐s t❤✉s σ(A) = R✳

slide-14
SLIDE 14

❋❡❛t✉r❡ ✭■✮✿ ❍✐❧❜❡rt s♣❛❝❡

■❞❡❛✿ ✉s❡ t❤❡ ❍✐❧❜❡rt s♣❛❝❡ ♠♦❞❡❧ ❢♦r s❡♠✐❜♦✉♥❞❡❞ ♦♣❡r❛t♦rs ❀ ❞❡❝♦♠♣♦s❡ A = A+ ⊕ −A− σ(A+) = [✵, ∞), σ(A−) = (✵, ∞) ❀ ❞❡❝♦♠♣♦s✐t✐♦♥ Hk(A) = H+

k (A+) ⊕ H− k (A−) ❛♥❞ ϕ = ϕ+ + ϕ−

■♥❣r❡❞✐❡♥ts ✭❛s ❜❡❢♦r❡✮✿

◮ µ✶, . . . , µn−✷, µ ∈ R✱ ❞✐st✐♥❝t✱ t♦ t❤❡ ❧❡❢t ♦❢ t❤❡ σ(A±) ◮ ♣♦❧②♥♦♠✐❛❧ b(z) = n−✷ i=✶ (z − µi)

❀ b(z) > ✵ ♦♥ σ(A±)

◮ ❞✐❛❣♦♥❛❧ ♠❛tr✐① M = ❞✐❛❣(µ✶, . . . , µn−✷) ◮ ❝♦❡✣❝✐❡♥ts ai ❢r♦♠ t❤❡ ♣❛rt✐❛❧ ❢r❛❝t✐♦♥ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ b(z)−✶ ◮ ❡❧❡♠❡♥ts g ±(µi) := (A± − µi)−✶ϕ±

∈ H±

−n+✷ ◮ t✇♦ ✈❡r② r❡❣✉❧❛r ❡❧❡♠❡♥ts

G ±(µ) := (A± − µ)−✶ n−✷

i=✶ (A± − µi)−✶ϕ

∈ Hn−✷

◮ t✇♦ ♣♦s✐t✐✈❡ ❞❡✜♥✐t❡ s✳❛✳ ♠❛tr✐❝❡s Γ± s✉❝❤ t❤❛t Γ±M = MΓ±

slide-15
SLIDE 15

❋❡❛t✉r❡ ✭■✮✿ ❍✐❧❜❡rt s♣❛❝❡ ■■

❚❤✐s ❛❧❧♦✇s ✉s t♦ ❜✉✐❧❞ H± := H±

n−✷ ⊕ Cn−✷

s❝✳♣✳ ❢r♦♠ b(A±) ⊕ Γ± ❢♦r t❤❡ ♠♦❞❡❧ s♣❛❝❡ s❡t H = H+ ⊕ H− ֒ → H−n+✷(A) ❉❡✜♥❡ t❤❡ ♠❛①✐♠❛❧ ♦♣❡r❛t♦r A♠❛① ❛♥❞ ✐ts ❛❞❥♦✐♥t A♠✐♥✿ ❞♦♠❛✐♥s ❛♥❞ ❛❝t✐♦♥s ❛r❡ s✐♠✐❧❛r t♦ ❜❡❢♦r❡ ✭❛❞❞ ± ❡✈❡r②✇❤❡r❡✮✱ ❡✳❣✳ A♠❛①        U+

r + u+G +(µ)

  • u+

U−

r + u−G −(µ)

  • u−

       =        A+U+

r + µu+G +(µ)

M u+ + u+ a −A−U−

r − µu−G −(µ)

−M u− − u− a        A♠✐♥ ✐s s②♠♠❡tr✐❝ ✇✐t❤ ❞❡❢❡❝t (✷, ✷)✿ ❀ t✇♦ ❞❡❢❡❝t ❡❧❡♠❡♥ts Φ±(z)✱ ❧❡t Φ(z) ❜❡ t❤❡✐r s✉♠✱ ❛♥❞ ❛ ✷ × ✷ Q✲❢✉♥❝t✐♦♥ Q ✲ ✐♥ ❢❛❝t✱ Q(z) = ❞✐❛❣(Q+(z), Q−(z))

slide-16
SLIDE 16

❋❡❛t✉r❡ ✭■✮✿ ❍✐❧❜❡rt s♣❛❝❡ ■■

❚❤✐s ❛❧❧♦✇s ✉s t♦ ❜✉✐❧❞ H± := H±

n−✷ ⊕ Cn−✷

s❝✳♣✳ ❢r♦♠ b(A±) ⊕ Γ± ❢♦r t❤❡ ♠♦❞❡❧ s♣❛❝❡ s❡t H = H+ ⊕ H− ֒ → H−n+✷(A) ❉❡✜♥❡ t❤❡ ♠❛①✐♠❛❧ ♦♣❡r❛t♦r A♠❛① ❛♥❞ ✐ts ❛❞❥♦✐♥t A♠✐♥✿ ❞♦♠❛✐♥s ❛♥❞ ❛❝t✐♦♥s ❛r❡ s✐♠✐❧❛r t♦ ❜❡❢♦r❡ ✭❛❞❞ ± ❡✈❡r②✇❤❡r❡✮✱ ❡✳❣✳ A♠❛①        U+

r + u+G +(µ)

  • u+

U−

r + u−G −(µ)

  • u−

       =        A+U+

r + µu+G +(µ)

M u+ + u+ a −A−U−

r − µu−G −(µ)

−M u− − u− a        A♠✐♥ ✐s s②♠♠❡tr✐❝ ✇✐t❤ ❞❡❢❡❝t (✷, ✷)✿ ❀ t✇♦ ❞❡❢❡❝t ❡❧❡♠❡♥ts Φ±(z)✱ ❧❡t Φ(z) ❜❡ t❤❡✐r s✉♠✱ ❛♥❞ ❛ ✷ × ✷ Q✲❢✉♥❝t✐♦♥ Q ✲ ✐♥ ❢❛❝t✱ Q(z) = ❞✐❛❣(Q+(z), Q−(z))

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SLIDE 17

❋❡❛t✉r❡ ✭■✮✿ ❍✐❧❜❡rt s♣❛❝❡ ■■■

❯s❡ A✵ := A+ ⊕ M ⊕ −A− ⊕ −M ❛s r❡❢❡r❡♥❝❡ ❜② ❑r❡✐♥✬s ❢♦r♠✉❧❛ ✇❡ ❝❛♥ ♣❛r❛♠❡tr✐s❡ ❛ ✭s✉❜✲✮❢❛♠✐❧② ♦❢ s✳❛✳ ♦♣❡r❛t♦rs t❤❛t ♠♦❞❡❧ t❤❡ ♣❡rt✉r❜❛t✐♦♥s✱ ✈✐❛ (Aβ − z)−✶ = (A✵ − z)−✶ − Φ(z), · Q+(z) + Q−(z) + ✶

β

Φ(z) ❇♦♥✉s✿ ❉❡s❝r✐♣t✐♦♥ ✈✐❛ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ❍♦✇❡✈❡r✱ ❛❢t❡r ❡♠❜❡❞❞✐♥❣✿ ι(Aβ − z)−✶|Hn−✷ = (A − z)−✶ − (A − z)−✶ϕ, · Q+(z) + Q−(z) + ✶

β

× × ✶ b(z)(A+ − z)−✶ϕ+ + ✶ b(−z)(−A− − z)−✶ϕ−

  • ❲❡ ❣❡t ❛♥ ❛❞❞✐t✐♦♥❛❧ t✇✐st ✐♥ t❤❡ ❝♦❡✣❝✐❡♥ts b(±z)−✶ ❀ ❋❡❛t✉r❡ ✭■■✮ ✐s ♥♦t

♣r❡s❡r✈❡❞✦

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SLIDE 18

❋❡❛t✉r❡ ✭■✮✿ ❍✐❧❜❡rt s♣❛❝❡ ■■■

❯s❡ A✵ := A+ ⊕ M ⊕ −A− ⊕ −M ❛s r❡❢❡r❡♥❝❡ ❜② ❑r❡✐♥✬s ❢♦r♠✉❧❛ ✇❡ ❝❛♥ ♣❛r❛♠❡tr✐s❡ ❛ ✭s✉❜✲✮❢❛♠✐❧② ♦❢ s✳❛✳ ♦♣❡r❛t♦rs t❤❛t ♠♦❞❡❧ t❤❡ ♣❡rt✉r❜❛t✐♦♥s✱ ✈✐❛ (Aβ − z)−✶ = (A✵ − z)−✶ − Φ(z), · Q+(z) + Q−(z) + ✶

β

Φ(z) ❇♦♥✉s✿ ❉❡s❝r✐♣t✐♦♥ ✈✐❛ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ❍♦✇❡✈❡r✱ ❛❢t❡r ❡♠❜❡❞❞✐♥❣✿ ι(Aβ − z)−✶|Hn−✷ = (A − z)−✶ − (A − z)−✶ϕ, · Q+(z) + Q−(z) + ✶

β

× × ✶ b(z)(A+ − z)−✶ϕ+ + ✶ b(−z)(−A− − z)−✶ϕ−

  • ❲❡ ❣❡t ❛♥ ❛❞❞✐t✐♦♥❛❧ t✇✐st ✐♥ t❤❡ ❝♦❡✣❝✐❡♥ts b(±z)−✶ ❀ ❋❡❛t✉r❡ ✭■■✮ ✐s ♥♦t

♣r❡s❡r✈❡❞✦

slide-19
SLIDE 19

❋❡❛t✉r❡ ✭■■✮✿ ❑r❡✐♥✬s ❢♦r♠✉❧❛

❆ss✉♠❡ ϕ ∈ H−✷k ❢♦r k ≥ ✷✳ ❲❡ ✇♦r❦ ❛❣❛✐♥ ❞✐r❡❝t❧② ✇✐t❤ A ❛♥❞ Hk(A)✳ ■♥❣r❡❞✐❡♥ts✿

◮ ν✶, . . . , νk−✶, ν∈ C+ ❛♥❞ νk = ν✶, . . . , ν✷k−✷ = νk−✶ ◮ ♣♦❧②♥♦♠✐❛❧ d(z) = k−✶ i=✶ (z − νi)(z − νk−✶+i)

❀ d(z) > ✵ ♦♥ σ(A)

◮ ❞✐❛❣♦♥❛❧ ♠❛tr✐① N = ❞✐❛❣(ν✶, . . . , ν✷k−✷) ◮ ❝♦❡✣❝✐❡♥ts ci ❢r♦♠ t❤❡ ♣❛rt✐❛❧ ❢r❛❝t✐♦♥ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ d(z)−✶ ◮ ❡❧❡♠❡♥ts g(νi) := (A − νi)−✶ϕ

∈ H−✷k+✷

◮ ♦♥❡ ✈❡r② r❡❣✉❧❛r ❡❧❡♠❡♥t

G(ν) := (A − ν)−✶ ✷k−✷

i=✶ (A − νi)−✶ϕ

∈ H✷k−✷

◮ ❛ s✳❛✳ ♠❛tr✐① Γ

✇✐t❤ t❤✐s ❜✉✐❧❞ ❛ s♣❛❝❡ Πk−✶ := H✷k−✷ ⊕ C✷k−✷✱ s❝✳♣✳ ❢r♦♠ d(A) ⊕ Γ

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SLIDE 20

❋❡❛t✉r❡ ✭■■✮✿ ❑r❡✐♥✬s ❢♦r♠✉❧❛ ■■

❢♦❧❧♦✇ t❤❡ s❛♠❡ ♣r♦❝❡❞✉r❡✿ ❞❡✜♥❡ A♠❛①✱ Amin = A∗

♠❛①

❀ A♠✐♥ ✐s s②♠♠❡tr✐❝ ✐✛ ΓN = N∗Γ ❀ ❝❛❧❝✉❧❛t❡ Q✲❢✉♥❝t✐♦♥ QΠ(z)✱ ❞❡❢❡❝t (✶, ✶)✱ ❞❡❢❡❝t ❡❧❡♠❡♥t Ψ(z)✱ s✳❛✳ ♦♣❡r❛t♦r A✵ = A ⊕ N ❛s r❡❢❡r❡♥❝❡ t♦ ❡♥❞ ✉♣ ✇✐t❤ ❋❡❛t✉r❡ ✭■■✮✿ ιΠ(Ah − z)−✶|H✷k−✷ = (A − z)−✶ − (A − z)−✶ϕ, · QΠ(z) + ✶

h

✶ d(z)(A − z)−✶ϕ ❍♦✇❡✈❡r✿ ❢♦r t❤❡ ❣✐✈❡♥ N ✇❡ ✜♥❞ t❤❛t ΓN = N∗Γ ✐✛ Γ =   ✵ Ξ Ξ∗ ✵   Ξ = ❞✐❛❣(ξ✶, . . . , ξk−✶), ξi ∈ C ❀ Γ ❤❛s k − ✶ ♥❡❣❛t✐✈❡ sq✉❛r❡s✱ Πk−✶ ✐s ❛ P♦♥tr②❛❣✐♥ s♣❛❝❡✦ ❀ ❋❡❛t✉r❡ ✭■✮ ❣♦❡s ♦✉t t❤❡ ✇✐♥❞♦✇✦

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SLIDE 21

❋❡❛t✉r❡ ✭■■✮✿ ❑r❡✐♥✬s ❢♦r♠✉❧❛ ■■

❢♦❧❧♦✇ t❤❡ s❛♠❡ ♣r♦❝❡❞✉r❡✿ ❞❡✜♥❡ A♠❛①✱ Amin = A∗

♠❛①

❀ A♠✐♥ ✐s s②♠♠❡tr✐❝ ✐✛ ΓN = N∗Γ ❀ ❝❛❧❝✉❧❛t❡ Q✲❢✉♥❝t✐♦♥ QΠ(z)✱ ❞❡❢❡❝t (✶, ✶)✱ ❞❡❢❡❝t ❡❧❡♠❡♥t Ψ(z)✱ s✳❛✳ ♦♣❡r❛t♦r A✵ = A ⊕ N ❛s r❡❢❡r❡♥❝❡ t♦ ❡♥❞ ✉♣ ✇✐t❤ ❋❡❛t✉r❡ ✭■■✮✿ ιΠ(Ah − z)−✶|H✷k−✷ = (A − z)−✶ − (A − z)−✶ϕ, · QΠ(z) + ✶

h

✶ d(z)(A − z)−✶ϕ ❍♦✇❡✈❡r✿ ❢♦r t❤❡ ❣✐✈❡♥ N ✇❡ ✜♥❞ t❤❛t ΓN = N∗Γ ✐✛ Γ =   ✵ Ξ Ξ∗ ✵   Ξ = ❞✐❛❣(ξ✶, . . . , ξk−✶), ξi ∈ C ❀ Γ ❤❛s k − ✶ ♥❡❣❛t✐✈❡ sq✉❛r❡s✱ Πk−✶ ✐s ❛ P♦♥tr②❛❣✐♥ s♣❛❝❡✦ ❀ ❋❡❛t✉r❡ ✭■✮ ❣♦❡s ♦✉t t❤❡ ✇✐♥❞♦✇✦

slide-22
SLIDE 22

❈♦♥❝❧✉s✐♦♥

❲❡ s❡❡ t❤❛t ❢♦r ♥♦♥✲s❡♠✐❜♦✉♥❞❡❞ s✳❛✳ ♦♣❡r❛t♦rs t❤❡ s✐t✉❛t✐♦♥ s♣❧✐ts✿

◮ ❊✐t❤❡r ②♦✉ ❜✉✐❧❞ ❛ ❍✐❧❜❡rt s♣❛❝❡✱ ❛t t❤❡ ❝♦st ♦❢ ✐♥tr♦❞✉❝✐♥❣ ❛ t✇✐st ✐♥

❑r❡✐♥✬s ❢♦r♠✉❧❛✱

◮ ❖r ②♦✉ ❦❡❡♣ ❑r❡✐♥✬s ❢♦r♠✉❧❛ ✐♥t❛❝t ❛t t❤❡ ❝♦st ♦❢ ❧✐✈✐♥❣ ✐♥ ❛ P♦♥tr②❛❣✐♥

s♣❛❝❡✳

❚❤❛♥❦ ②♦✉ ❢♦r ②♦✉r ❛tt❡♥t✐♦♥✦

slide-23
SLIDE 23

❈♦♥❝❧✉s✐♦♥

❲❡ s❡❡ t❤❛t ❢♦r ♥♦♥✲s❡♠✐❜♦✉♥❞❡❞ s✳❛✳ ♦♣❡r❛t♦rs t❤❡ s✐t✉❛t✐♦♥ s♣❧✐ts✿

◮ ❊✐t❤❡r ②♦✉ ❜✉✐❧❞ ❛ ❍✐❧❜❡rt s♣❛❝❡✱ ❛t t❤❡ ❝♦st ♦❢ ✐♥tr♦❞✉❝✐♥❣ ❛ t✇✐st ✐♥

❑r❡✐♥✬s ❢♦r♠✉❧❛✱

◮ ❖r ②♦✉ ❦❡❡♣ ❑r❡✐♥✬s ❢♦r♠✉❧❛ ✐♥t❛❝t ❛t t❤❡ ❝♦st ♦❢ ❧✐✈✐♥❣ ✐♥ ❛ P♦♥tr②❛❣✐♥

s♣❛❝❡✳

❚❤❛♥❦ ②♦✉ ❢♦r ②♦✉r ❛tt❡♥t✐♦♥✦