stt r - - PowerPoint PPT Presentation

❆s②♠♣t♦t✐❝ ❇❡❤❛✈✐♦✉r ❛♥❞ ❈②❝❧✐❝ Pr♦♣❡rt✐❡s ♦❢ ❚r❡❡✲s❤✐❢t ❖♣❡r❛t♦rs

• ②ör❣② Pá❧ ●❡❤ér

❯♥✐✈❡rs✐t② ♦❢ ❙③❡❣❡❞✱ ❇♦❧②❛✐ ■♥st✐t✉t❡ ✷✾t❤ ❏✉❧② ✷✵✶✸✱ ●♦t❤❡♥❜✉r❣ ❇❛♥❛❝❤ ❆❧❣❡❜r❛s ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s

✶

■♥tr♦❞✉❝t✐♦♥✱ ▼♦t✐✈❛t✐♦♥ ❉✐r❡❝t❡❞ ❚r❡❡s ❛♥❞ ❚r❡❡s❤✐❢t ❖♣❡r❛t♦rs ❆s②♠♣t♦t✐❝ ❇❡❤❛✈✐♦✉r ♦❢ ❍✐❧❜❡rt ❙♣❛❝❡ ❈♦♥tr❛❝t✐♦♥s

✷

❆s②♠♣t♦t✐❝ ▲✐♠✐ts ❛♥❞ ■s♦♠❡tr✐❝ ❆s②♠♣t♦t❡s ♦❢ ❚r❡❡✲s❤✐❢t ❖♣❡r❛t♦rs

✸

❈②❝❧✐❝ Pr♦♣❡rt✐❡s ❈②❝❧✐❝✐t② ♦❢ ❚r❡❡✲s❤✐❢t ❖♣❡r❛t♦rs ❙✐♠✐❧❛r✐t② t♦ ♦rt❤♦❣♦♥❛❧ s✉♠ ♦❢ ❜✐✲ ❛♥❞ ✉♥✐❧❛t❡r❛❧ s❤✐❢ts ❈②❝❧✐❝✐t② ♦❢ t❤❡ ❆❞❥♦✐♥t

❆s②♠♣t♦t✐❝ ❇❡❤❛✈✐♦✉r ❛♥❞ ❈②❝❧✐❝ Pr♦♣❡rt✐❡s ♦❢ ❚r❡❡✲s❤✐❢t ❖♣❡r❛t♦rs ■♥tr♦❞✉❝t✐♦♥✱ ▼♦t✐✈❛t✐♦♥

❲❡✐❣❤t❡❞ ✉♥✐❧❛t❡r❛❧ s❤✐❢ts✿ {❡♥}∞

♥=✵ ✐s ❛♥ ❖◆❇ ✐♥ H✱ {✇♥}∞ ♥=✶ ⊆ C

❜♦✉♥❞❡❞ ✇❡✐❣❤t✲s❡q✉❡♥❝❡ ❲ : H → H, ❲❡♥ = ✇♥+✶❡♥+✶. ❲❡✐❣❤t❡❞ ❜✐❧❛t❡r❛❧ s❤✐❢ts✿ {❡♥}∞

♥=−∞ ✐s ❛♥ ❖◆❇ ✐♥ H✱

{✇♥}∞

♥=−∞ ⊆ C ❜♦✉♥❞❡❞ ✇❡✐❣❤t✲s❡q✉❡♥❝❡

❲ : H → H, ❲❡♥ = ✇♥+✶❡♥+✶. ❲❡✐❣❤t❡❞ ❜❛❝❦✇❛r❞ s❤✐❢ts✿ {❡♥}∞

♥=✵ ✐s ❛♥ ❖◆❇ ✐♥ H✱ {✇♥}∞ ♥=✵ ⊆ C

❜♦✉♥❞❡❞ ✇❡✐❣❤t✲s❡q✉❡♥❝❡ ❲ : H → H, ❲❡♥ = ✇♥−✶❡♥−✶ ✐❢ ♥ > ✵, ✵ ✐❢ ♥ = ✵.

• ②ör❣② Pá❧ ●❡❤ér

❆s②♠♣t♦t✐❝ ❇❡❤❛✈✐♦✉r ❛♥❞ ❈②❝❧✐❝ Pr♦♣❡rt✐❡s ♦❢ ❚r❡❡✲s❤✐❢t ❖♣❡r❛t♦rs

❆s②♠♣t♦t✐❝ ❇❡❤❛✈✐♦✉r ❛♥❞ ❈②❝❧✐❝ Pr♦♣❡rt✐❡s ♦❢ ❚r❡❡✲s❤✐❢t ❖♣❡r❛t♦rs ■♥tr♦❞✉❝t✐♦♥✱ ▼♦t✐✈❛t✐♦♥

• r❛♣❤s ❝❛♥ ❜❡ ❛ss♦❝✐❛t❡❞ t♦ t❤❡s❡ ♦♣❡r❛t♦rs✿

❙♦♠❡t✐♠❡s ✇❡ ✇r✐t❡ t❤❡ ✇❡✐❣❤ts ♦♥ t❤❡ ❡❞❣❡s✳

• ②ör❣② Pá❧ ●❡❤ér

❆s②♠♣t♦t✐❝ ❇❡❤❛✈✐♦✉r ❛♥❞ ❈②❝❧✐❝ Pr♦♣❡rt✐❡s ♦❢ ❚r❡❡✲s❤✐❢t ❖♣❡r❛t♦rs

❆s②♠♣t♦t✐❝ ❇❡❤❛✈✐♦✉r ❛♥❞ ❈②❝❧✐❝ Pr♦♣❡rt✐❡s ♦❢ ❚r❡❡✲s❤✐❢t ❖♣❡r❛t♦rs ■♥tr♦❞✉❝t✐♦♥✱ ▼♦t✐✈❛t✐♦♥

❲❤② ✇❡ ✐♥✈❡st✐❣❛t❡ s❤✐❢t ♦♣❡r❛t♦rs❄ ❙✐♠♣❧❡ str✉❝t✉r❡✳ ❚❤❡② ♣r♦✈✐❞❡ ❡①❛♠♣❧❡s ❢♦r ♠❛♥② q✉❡st✐♦♥s✳ ❆❢t❡r ♥♦r♠❛❧ ♦♣❡r❛t♦rs t❤❡② ❛r❡ t❤❡ ❜❡st t❡st ♦♣❡r❛t♦rs✳ ❆ ♥❛t✉r❛❧ ❣❡♥❡r❛❧✐③❛t✐♦♥✿ r❡♣❧❛❝❡ t❤❡ ♣r❡✈✐♦✉s ❣r❛♣❤s ✇✐t❤ ❞✐r❡❝t❡❞ tr❡❡s ❛♥❞ ✇r✐t❡ s♦♠❡ ✇❡✐❣❤ts ✐♥ t❤❡ ✈❡rt✐❝❡s ✇❤✐❝❤ ❛r❡ ♥♦t r♦♦ts✳ − → ❚❤✐s ✇✐❧❧ ♥❛t✉r❛❧❧② ❞❡✜♥❡ ❛ ✒tr❡❡✲s❤✐❢t ♦♣❡r❛t♦r✑✳ ❚❤❡② ✇❡r❡ r❡❝❡♥t❧② ✐♥tr♦❞✉❝❡❞ ✐♥✿ ❩✳ ❏✳ ❏❛❜❧♦♥s❦✐✱ ■✳ ❇✳ ❏✉♥❣ ❛♥❞ ❏✳ ❙t♦❝❤❡❧✱ ❲❡✐❣❤t❡❞ ❙❤✐❢ts ♦♥ ❉✐r❡❝t❡❞ ❚r❡❡s✱ ▼❡♠♦✐rs ♦❢ t❤❡ ❆♠❡r✐❝❛♥ ▼❛t❤❡♠❛t✐❝❛❧ ❙♦❝✐❡t②✱ ◆✉♠❜❡r ✶✵✶✼✱ ✷✵✶✷✳ ✭❤②♣♦♥♦r♠❛❧✐t②✱ ❝♦✲❤②♣♦♥♦r♠❛❧✐t②✱ s✉❜♥♦r♠❛❧✐t② ❛♥❞ ❤②♣❡r❡①♣❛♥s✐✈✐t② ✕ s✐♠♣❧❡r ❡①❛♠♣❧❡s t❤❛♥ ❜❡❢♦r❡✮

• ②ör❣② Pá❧ ●❡❤ér

❆s②♠♣t♦t✐❝ ❇❡❤❛✈✐♦✉r ❛♥❞ ❈②❝❧✐❝ Pr♦♣❡rt✐❡s ♦❢ ❚r❡❡✲s❤✐❢t ❖♣❡r❛t♦rs

❆s②♠♣t♦t✐❝ ❇❡❤❛✈✐♦✉r ❛♥❞ ❈②❝❧✐❝ Pr♦♣❡rt✐❡s ♦❢ ❚r❡❡✲s❤✐❢t ❖♣❡r❛t♦rs ■♥tr♦❞✉❝t✐♦♥✱ ▼♦t✐✈❛t✐♦♥ ❉✐r❡❝t❡❞ ❚r❡❡s ❛♥❞ ❚r❡❡s❤✐❢t ❖♣❡r❛t♦rs

T = (❱ , ❊) ✐s ❛ ❞✐r❡❝t❡❞ ❣r❛♣❤✱ ❱ ≡ ✈❡rt✐❝❡s✱ ❊ ≡ ✭❞✐r❡❝t❡❞✮ ❡❞❣❡s ❊ ⊆ ❱ × ❱ \ {(✈, ✈): ✈ ∈ ❱ }✳ T ✐s ❛ ❞✐r❡❝t❡❞ tr❡❡ ✐❢ ✭✐✮ T ✐s ❝♦♥♥❡❝t❡❞✱ ✐✳❡✳✿ ∀ ✉, ✈ ∈ ❱ , ✉ = ✈ ∃ ✉ = ✈✵, ✈✶, . . . ✈♥ = ✈ ∈ ❱ , ♥ ∈ N s✳ t✳ (✈❥−✶, ✈❥) ♦r (✈❥, ✈❥−✶) ∈ ❊ ❢♦r ❡✈❡r② ✶ ≤ ❥ ≤ ♥✳ ✭✐✐✮ ❋♦r ❡❛❝❤ ✈❡rt❡① ✈ t❤❡r❡ ❡①✐sts ❛t ♠♦st ♦♥❡ ♦t❤❡r ✈❡rt❡① ✉ ✇✐t❤ t❤❡ ♣r♦♣❡rt② t❤❛t (✉, ✈) ∈ ❊ ✭✐✳❡✳ ❡✈❡r② ✈❡rt❡① ❤❛s ❛t ♠♦st ♦♥❡ ♣❛r❡♥t✮✱ ❛♥❞ ✭✐✐✐✮ T ❤❛s ♥♦ ✭❞✐r❡❝t❡❞✮ ❝✐r❝✉✐t✱ ✐✳❡✳✿ ∄ ✈✵, ✈✶, . . . ✈♥ ∈ ❱ , ♥ ∈ N ❞✐st✐♥❝t ✈❡rt✐❝❡s s✳ t✳ (✈❥−✶, ✈❥) ∈ ❊ ∀ ✶ ≤ ❥ ≤ ♥ ❛♥❞ (✈♥, ✈✵) ∈ ❊✳ ■❢ (✉, ✈) ∈ ❊✱ t❤❡♥ ✈ ✐s ❛ ❝❤✐❧❞ ♦❢ ✉✱ ✉ ✐s t❤❡ ♣❛r❡♥t ♦❢ ✈✳ ■♥ ♥♦t❛t✐♦♥s✿ ✈ ∈ ❈❤✐(✉)✱ ♣❛rT (✈) = ♣❛r(✈) = ✉✳ ❈❤✐T (✉) = ❈❤✐(✉) ≡ t❤❡ s❡t ♦❢ ❛❧❧ ❝❤✐❧❞r❡♥ ♦❢ ✉✳ ■❢ ✈ ❤❛s ❛ ♣❛r❡♥t✱ t❤❡♥ ✐t ✐s ✉♥✐q✉❡✳

• ②ör❣② Pá❧ ●❡❤ér

❆s②♠♣t♦t✐❝ ❇❡❤❛✈✐♦✉r ❛♥❞ ❈②❝❧✐❝ Pr♦♣❡rt✐❡s ♦❢ ❚r❡❡✲s❤✐❢t ❖♣❡r❛t♦rs

❆s②♠♣t♦t✐❝ ❇❡❤❛✈✐♦✉r ❛♥❞ ❈②❝❧✐❝ Pr♦♣❡rt✐❡s ♦❢ ❚r❡❡✲s❤✐❢t ❖♣❡r❛t♦rs ■♥tr♦❞✉❝t✐♦♥✱ ▼♦t✐✈❛t✐♦♥ ❉✐r❡❝t❡❞ ❚r❡❡s ❛♥❞ ❚r❡❡s❤✐❢t ❖♣❡r❛t♦rs

♣❛r❦(✈) := ♣❛r(. . . (♣❛r

• ❦✲t✐♠❡s

(✈)) . . . ) ✐❢ ✐t ♠❛❦❡s s❡♥s❡✱ ❛♥❞ ♣❛r✵ ✐s t❤❡ ✐❞❡♥t✐t② ♠❛♣✳ ■❢ ✉ ❤❛s ♥♦ ♣❛r❡♥t − → r♦♦t✳ ■❢ t❤❡r❡ ❡①✐sts ♦♥❡✱ t❤❡♥ ✐t ✐s ✉♥✐q✉❡ ✉ := r♦♦t = r♦♦tT ✳ ■❢ ❈❤✐(✉) = ∅ − → ❧❡❛❢✳ ▲❡❛(T ) ≡ t❤❡ s❡t ♦❢ ❛❧❧ ❧❡❛✈❡s✳ ■❢ ❲ ⊆ ❱ ✱ t❤❡♥ ❈❤✐(❲ ) := ∪✈∈❲ ❈❤✐(✈)✱

• ❡♥T (✉) = ●❡♥(✉) = ∞

♥=✵

♥

❥=✵ ❈❤✐❥(♣❛r❥(✉))

• ✐s t❤❡

✭✇❤♦❧❡✮ ❣❡♥❡r❛t✐♦♥ ♦r t❤❡ ❧❡✈❡❧ ♦❢ ✉✳

• ②ör❣② Pá❧ ●❡❤ér

❆s②♠♣t♦t✐❝ ❇❡❤❛✈✐♦✉r ❛♥❞ ❈②❝❧✐❝ Pr♦♣❡rt✐❡s ♦❢ ❚r❡❡✲s❤✐❢t ❖♣❡r❛t♦rs

❆s②♠♣t♦t✐❝ ❇❡❤❛✈✐♦✉r ❛♥❞ ❈②❝❧✐❝ Pr♦♣❡rt✐❡s ♦❢ ❚r❡❡✲s❤✐❢t ❖♣❡r❛t♦rs ■♥tr♦❞✉❝t✐♦♥✱ ▼♦t✐✈❛t✐♦♥ ❉✐r❡❝t❡❞ ❚r❡❡s ❛♥❞ ❚r❡❡s❤✐❢t ❖♣❡r❛t♦rs

❇♦✉♥❞❡❞ ❚r❡❡✲s❤✐❢t ❖♣❡r❛t♦r

ℓ✷(❱ )✿ ❝♦♠♣❧❡① ❍✐❧❜❡rt s♣❛❝❡ ♦❢ ❛❧❧ sq✉❛r❡ s✉♠♠❛❜❧❡ ❢✉♥❝t✐♦♥s✳ ❚❤❡ ♥❛t✉r❛❧ ✐♥♥❡r ♣r♦❞✉❝t✿

• ❢ , ❣
• =

✉∈❱ ❢ (✉)❣(✉)✳

❡✉(✈) = δ✉,✈ ✭✉ ∈ ❱ ✮✱ {❡✉}✉∈❱ ❖◆❇✳ ❲ ⊆ ❱ − → ℓ✷(❲ ) = ∨{❡✈ : ✈ ∈ ❲ } ✇❤✐❝❤ ✐s ❛ s✉❜s♣❛❝❡ ✭❝❧♦s❡❞ ❧✐♥❡❛r ♠❛♥✐❢♦❧❞✮✳ ▲❡t λ = {λ✈ : ✈ ∈ ❱ \ {r♦♦t}} ⊆ C ✭✈ ∈ ❱ ✐❢ T ✐s r♦♦t❧❡ss✮ ❜❡ ❛ s❡t ♦❢ ✇❡✐❣❤ts s✉♣

• ✈∈❈❤✐(✉) |λ✈|✷ : ✉ ∈ ❱
• < ∞

❙λ : ℓ✷(❱) → ℓ✷(❱), ❡✉ →

• ✈∈❈❤✐(✉)

λ✈❡✈, ❙λ = s✉♣   

• ✈∈❈❤✐(✉)

|λ✈|✷ : ✉ ∈ ❱   

• ②ör❣② Pá❧ ●❡❤ér

❆s②♠♣t♦t✐❝ ❇❡❤❛✈✐♦✉r ❛♥❞ ❈②❝❧✐❝ Pr♦♣❡rt✐❡s ♦❢ ❚r❡❡✲s❤✐❢t ❖♣❡r❛t♦rs

❆s②♠♣t♦t✐❝ ❇❡❤❛✈✐♦✉r ❛♥❞ ❈②❝❧✐❝ Pr♦♣❡rt✐❡s ♦❢ ❚r❡❡✲s❤✐❢t ❖♣❡r❛t♦rs ■♥tr♦❞✉❝t✐♦♥✱ ▼♦t✐✈❛t✐♦♥ ❆s②♠♣t♦t✐❝ ❇❡❤❛✈✐♦✉r ♦❢ ❍✐❧❜❡rt ❙♣❛❝❡ ❈♦♥tr❛❝t✐♦♥s

H✿ ❝♦♠♣❧❡① ❍✐❧❜❡rt s♣❛❝❡✱ B(H)✿ ❜♦✉♥❞❡❞ ❧✐♥❡❛r ♦♣❡r❛t♦rs ♦♥ ✐t✳ ❙✉♣♣♦s❡ ❚ ∈ B(H) ✐s ❛ ❝♦♥tr❛❝t✐♦♥✿ ❚ ≤ ✶✱ t❤❡♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❙❖❚✲❧✐♠✐ts ❡①✐st✿ ❆ = ❆❚ = ❧✐♠

♥→∞ ❚ ∗♥❚ ♥

❛♥❞ ❆∗ = ❆❚ ∗ = ❧✐♠

♥→∞ ❚ ♥❚ ∗♥.

❆✿ t❤❡ ❛s②♠♣t♦t✐❝ ❧✐♠✐t ♦❢ ❚✱ ❆∗✿ t❤❡ ❛s②♠♣t♦t✐❝ ❧✐♠✐t ♦❢ ❚ ∗✳ ❤ ∈ H ✐s ❛ st❛❜❧❡ ✈❡❝t♦r ❢♦r ❚✱ ✐❢ ❧✐♠♥→∞ ❚ ♥❤ = ✵✳ ❚❤❡ s❡t ♦❢ ❛❧❧ st❛❜❧❡ ✈❡❝t♦rs✿ H✵ = H✵(❚) = N(❆❚)✳ H✵ ✐s ❛ ❤②♣❡r✐♥✈❛r✐❛♥t s✉❜s♣❛❝❡ ♦❢ ❚ ❛♥❞ ✇✐❧❧ ❜❡ ❝❛❧❧❡❞ t❤❡ st❛❜❧❡ s✉❜s♣❛❝❡ ♦❢ ❚✳ ✭❛ s✉❜s♣❛❝❡ ✐s ❤②♣❡r✐♥✈❛r✐❛♥t ❢♦r ❚ ✐❢ ✐t ✐s ✐♥✈❛r✐❛♥t ❢♦r ❡✈❡r② ❈ ❝♦♠♠✉t✐♥❣ ✇✐t❤ ❚✮

• ②ör❣② Pá❧ ●❡❤ér

❆s②♠♣t♦t✐❝ ❇❡❤❛✈✐♦✉r ❛♥❞ ❈②❝❧✐❝ Pr♦♣❡rt✐❡s ♦❢ ❚r❡❡✲s❤✐❢t ❖♣❡r❛t♦rs

❆s②♠♣t♦t✐❝ ❇❡❤❛✈✐♦✉r ❛♥❞ ❈②❝❧✐❝ Pr♦♣❡rt✐❡s ♦❢ ❚r❡❡✲s❤✐❢t ❖♣❡r❛t♦rs ■♥tr♦❞✉❝t✐♦♥✱ ▼♦t✐✈❛t✐♦♥ ❆s②♠♣t♦t✐❝ ❇❡❤❛✈✐♦✉r ♦❢ ❍✐❧❜❡rt ❙♣❛❝❡ ❈♦♥tr❛❝t✐♦♥s

❆ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❈♦♥tr❛❝t✐♦♥s

■❢ ❡✈❡r② ✈❡❝t♦r ✐s st❛❜❧❡✱ t❤❡♥ ❚ ✐s st❛❜❧❡ ♦r ♦❢ ❝❧❛ss ❈✵·✳ ■❢ H✵ = {✵}✱ t❤❡♥ ❚ ✐s ♦❢ ❝❧❛ss ❈✶·✳ ■❢ ❚ ∗ ∈ ❈✐·(H) ✭✐ = ✵ ♦r ✶✮✱ t❤❡♥ ❚ ✐s ♦❢ ❝❧❛ss ❈·✐✳ ❈✐❥(H) = ❈✐·(H) ∩ ❈·❥(H)✳ ❇❡❝❛✉s❡ H✵(❚) ♦r H✵(❚ ∗)⊥ ✐s ❤②♣❡r✐♥✈❛r✐❛♥t✱ ✐❢ ❚ / ∈ ❈✵✵ ∪ ❈✶✵ ∪ ❈✵✶ ∪ ❈✶✶✱ t❤❡♥ ✐t ❤❛s ❛ ♥♦♥✲tr✐✈✐❛❧ ❤②♣❡r✐♥✈❛r✐❛♥t s✉❜s♣❛❝❡ ✭❈✶✶ ❝❛♥ ❜❡ ❛❧s♦ ❤❛♥❞❧❡❞✮✳ ❋♦r ❈✶✶ ❝♦♥tr❛❝t✐♦♥s t❤❡r❡ ✐s ❛ ♥✐❝❡ str✉❝t✉r❡ r❡s✉❧t ✭❈✳ ❋♦✐❛s ❛♥❞ ❇✳ ❙③✳✲◆❛❣② ✮ ❙♦ t❤❡ ❤②♣❡r✲✐♥✈❛r✐❛♥t s✉❜s♣❛❝❡ ♣r♦❜❧❡♠ ✐s ♦♥❧② ♦♣❡♥ ✐♥ t❤❡ ❝❧❛ss❡s ❈✵✵, ❈✶✵, ❈✵✶✳

• ②ör❣② Pá❧ ●❡❤ér

❆s②♠♣t♦t✐❝ ❇❡❤❛✈✐♦✉r ❛♥❞ ❈②❝❧✐❝ Pr♦♣❡rt✐❡s ♦❢ ❚r❡❡✲s❤✐❢t ❖♣❡r❛t♦rs

❆s②♠♣t♦t✐❝ ❇❡❤❛✈✐♦✉r ❛♥❞ ❈②❝❧✐❝ Pr♦♣❡rt✐❡s ♦❢ ❚r❡❡✲s❤✐❢t ❖♣❡r❛t♦rs ■♥tr♦❞✉❝t✐♦♥✱ ▼♦t✐✈❛t✐♦♥ ❆s②♠♣t♦t✐❝ ❇❡❤❛✈✐♦✉r ♦❢ ❍✐❧❜❡rt ❙♣❛❝❡ ❈♦♥tr❛❝t✐♦♥s

❳ ∈ B(H, R(❆❚)−), ❳❤ = ❆✶/✷

❚ ❤✳

❚❤❡r❡ ❡①✐sts ❛ ✉♥✐q✉❡ ✐s♦♠❡tr② ❯ ∈ B(R(❆❚)−) s✳ t✳ ❳❚ = ❯❳. ❚❤❡ ♣❛✐r (❳, ❯) ✐s ❛ ❝❛♥♦♥✐❝❛❧ r❡❛❧✐③❛t✐♦♥ ♦❢ t❤❡ s♦ ❝❛❧❧❡❞ ✐s♦♠❡tr✐❝ ❛s②♠♣t♦t❡ ♦❢ ❚✳ ✭❚❤❡r❡ ✐s ❛ ♠♦r❡ ❣❡♥❡r❛❧ ❞❡✜♥✐t✐♦♥✱ ❜✉t ✇❡ ✇✐❧❧ ♥♦t ✉s❡ ✐t✮ ✭■♠♣♦rt❛♥t ❜❡❝❛✉s❡ ♦❢ t❤❡ ✭❤②♣❡r✮✐♥✈❛r✐❛♥t s✉❜s♣❛❝❡ ♣r♦❜❧❡♠✮ ❆♣♣❧✐❝❛t✐♦♥ ✶ ✭❈✳ ❋♦✐❛s ❛♥❞ ❇✳ ❙③✳✲◆❛❣②✮✿ ❲✐t❤ t❤❡ ✐s♦♠❡tr✐❝ ❛s②♠♣t♦t❡ ✇❡ ❝❛♥ ♣r♦✈❡ t❤❛t ❛ ❝♦♥tr❛❝t✐♦♥ ❚ ❢r♦♠ t❤❡ ❝❧❛ss ❈✶✶ ❛r❡ q✉❛s✐✲s✐♠✐❧❛r t♦ ❛ ✉♥✐t❛r② ♦♣❡r❛t♦r✳ ❚ ∼ ❯ ✭q✉❛s✐✲s✐♠✐❧❛r✮✱ ✐❢ ∃ ❳, ❨ ∈ B(H) ✇✐t❤ ❞❡♥s❡ r❛♥❣❡ ❛♥❞ tr✐✈✐❛❧ ❦❡r♥❡❧ s✳ t✳ ❳❚ = ❯❳ ❛♥❞ ❨❯ = ❚❨ ✳ ✭■❢ t❤❡② ❛r❡ s✐♠✐❧❛r✱ t❤❡♥ ❨ = ❳ −✶✮✳

• ②ör❣② Pá❧ ●❡❤ér

❆s②♠♣t♦t✐❝ ❇❡❤❛✈✐♦✉r ❛♥❞ ❈②❝❧✐❝ Pr♦♣❡rt✐❡s ♦❢ ❚r❡❡✲s❤✐❢t ❖♣❡r❛t♦rs

❆s②♠♣t♦t✐❝ ❇❡❤❛✈✐♦✉r ❛♥❞ ❈②❝❧✐❝ Pr♦♣❡rt✐❡s ♦❢ ❚r❡❡✲s❤✐❢t ❖♣❡r❛t♦rs ■♥tr♦❞✉❝t✐♦♥✱ ▼♦t✐✈❛t✐♦♥ ❆s②♠♣t♦t✐❝ ❇❡❤❛✈✐♦✉r ♦❢ ❍✐❧❜❡rt ❙♣❛❝❡ ❈♦♥tr❛❝t✐♦♥s

❤ ∈ H ✐s ❛ ❝②❝❧✐❝ ✈❡❝t♦r ❢♦r ❚ ∈ B(H) ∨{❚ ♥❤: ♥ ∈ Z+} = H. ❚❤❡♥ ❚ ✐s ❝②❝❧✐❝ ♦♣❡r❛t♦r✳ ❆♣♣❧✐❝❛t✐♦♥ ✷✿ ■❢ ❯ ✐s ♥♦t ❝②❝❧✐❝ ❛♥❞ ❚ ∈ ❈✶·(H) = ⇒ ❚ ✐s ❛❧s♦ ♥♦♥✲❝②❝❧✐❝✳ ■❢ ❯∗ ✐s ❝②❝❧✐❝ ❛♥❞ ❚ ∈ ❈✶·(H) = ⇒ ❚ ∗ ✐s ❛❧s♦ ❝②❝❧✐❝✳ ❆ ❝♦♥s❡q✉❡♥❝❡✿ ❊✈❡r② ❝♦♥tr❛❝t✐✈❡ ❈·✶ ✇❡✐❣❤t❡❞ ❜✐❧❛t❡r❛❧ s❤✐❢t ✐s ❝②❝❧✐❝✳ ❆♣♣❧✐❝❛t✐♦♥ ✸✿ ❚ ✐s s✐♠✐❧❛r t♦ ❛♥ ✐s♦♠❡tr② ⇐ ⇒ ❆❚ ✐s ✐♥✈❡rt✐❜❧❡✳ ❚ ✐s s✐♠✐❧❛r t♦ ❛ ✉♥✐t❛r② ⇐ ⇒ ❜♦t❤ ❆ ❛♥❞ ❆∗ ❛r❡ ✐♥✈❡rt✐❜❧❡✳

• ②ör❣② Pá❧ ●❡❤ér

❆s②♠♣t♦t✐❝ ❇❡❤❛✈✐♦✉r ❛♥❞ ❈②❝❧✐❝ Pr♦♣❡rt✐❡s ♦❢ ❚r❡❡✲s❤✐❢t ❖♣❡r❛t♦rs

❆s②♠♣t♦t✐❝ ❇❡❤❛✈✐♦✉r ❛♥❞ ❈②❝❧✐❝ Pr♦♣❡rt✐❡s ♦❢ ❚r❡❡✲s❤✐❢t ❖♣❡r❛t♦rs ❆s②♠♣t♦t✐❝ ▲✐♠✐ts ❛♥❞ ■s♦♠❡tr✐❝ ❆s②♠♣t♦t❡s ♦❢ ❚r❡❡✲s❤✐❢t ❖♣❡r❛t♦rs

❲✳▲✳♦✳●✳ ✇❡ ❝❛♥ ❛♥❞ ✇❡ ✇✐❧❧ ❛ss✉♠❡ ✐♥ t❤❡ t❛❧❦ t❤❛t ❛❧❧ ✇❡✐❣❤ts ❛r❡ str✐❝t❧② ♣♦s✐t✐✈❡ ❛♥❞ H ✐s s❡♣❛r❛❜❧❡ ✭|❱ | ≤ ℵ✵✮✳ ❚❤❡♦r❡♠ ▲❡t ❙λ ❜❡ ❛ tr❡❡✲s❤✐❢t ❝♦♥tr❛❝t✐♦♥✳ ❚❤❡♥ t❤❡ ❛s②♠♣t♦t✐❝ ❧✐♠✐t ❆ ✐s ❛ ♣♦s✐t✐✈❡ ♦♣❡r❛t♦r s✳ t✳ ❆❡✉ = α✉❡✉ ∀ ✉ ∈ ❱ , ✇❤❡r❡ α✉ = ❧✐♠♥→∞

• ✈∈❈❤✐♥(✉)

♥−✶

❥=✵ λ✷ ♣❛r❥(✈)✳

• ②ör❣② Pá❧ ●❡❤ér

❆s②♠♣t♦t✐❝ ❇❡❤❛✈✐♦✉r ❛♥❞ ❈②❝❧✐❝ Pr♦♣❡rt✐❡s ♦❢ ❚r❡❡✲s❤✐❢t ❖♣❡r❛t♦rs

❆s②♠♣t♦t✐❝ ❇❡❤❛✈✐♦✉r ❛♥❞ ❈②❝❧✐❝ Pr♦♣❡rt✐❡s ♦❢ ❚r❡❡✲s❤✐❢t ❖♣❡r❛t♦rs ❆s②♠♣t♦t✐❝ ▲✐♠✐ts ❛♥❞ ■s♦♠❡tr✐❝ ❆s②♠♣t♦t❡s ♦❢ ❚r❡❡✲s❤✐❢t ❖♣❡r❛t♦rs

❈♦r♦❧❧❛r② ■❢ ❱ ′ = {✈′ ∈ ❱ : ❡✈′ / ∈ H✵}✱ t❤❡♥ H✵ = ℓ✷(❱ \ ❱ ′) ❛♥❞ H⊥

✵ = ℓ✷(❱ ′)✳

Pr♦♣♦s✐t✐♦♥ ❚❤❡ ❢♦❧❧♦✇✐♥❣s ❛r❡ ✈❛❧✐❞ ❢♦r ❡✈❡r② ✈❡rt❡① ✉ ∈ ❱ ✿ ✭✐✮ ✉ / ∈ ❱ ′ = ⇒ ❈❤✐(✉) ⊆ ❱ \ ❱ ′✱ ✭✐✐✮ ✉ / ∈ ❱ ′ ⇐ ⇒ ❈❤✐(✉) ⊆ ❱ \ ❱ ′❀ t❤✐s ✐s ❢✉❧✜❧❧❡❞ ✐♥ t❤❡ s♣❡❝✐❛❧ ❝❛s❡ ✇❤❡♥ ✉ ✐s ❛ ❧❡❛❢✱ ✭✐✐✐✮ ✉ ∈ ❱ ′ = ⇒ ♣❛r❦(✉) ∈ ❱ ′, ∀ ❦ ∈ Z+✱ ✭✐✈✮ T ′ = (❱ ′, ❊ ′ = ❊ ∩ (❱ ′ × ❱ ′)) ✐s ❛ ❧❡❛✢❡ss s✉❜tr❡❡✱ ✭✈✮ ✐❢ T ❤❛s ♥♦ r♦♦t✱ ♥❡✐t❤❡r ❤❛s T ′✱ ❛♥❞ ✭✈✐✮ ✐❢ T ❤❛s ❛ r♦♦t✱ t❤❡♥ ❡✐t❤❡r ❙λ ∈ ❈✵·(ℓ✷(❱ )) ♦r r♦♦tT = r♦♦tT ′✳

• ②ör❣② Pá❧ ●❡❤ér

❆s②♠♣t♦t✐❝ ❇❡❤❛✈✐♦✉r ❛♥❞ ❈②❝❧✐❝ Pr♦♣❡rt✐❡s ♦❢ ❚r❡❡✲s❤✐❢t ❖♣❡r❛t♦rs

❆s②♠♣t♦t✐❝ ❇❡❤❛✈✐♦✉r ❛♥❞ ❈②❝❧✐❝ Pr♦♣❡rt✐❡s ♦❢ ❚r❡❡✲s❤✐❢t ❖♣❡r❛t♦rs ❆s②♠♣t♦t✐❝ ▲✐♠✐ts ❛♥❞ ■s♦♠❡tr✐❝ ❆s②♠♣t♦t❡s ♦❢ ❚r❡❡✲s❤✐❢t ❖♣❡r❛t♦rs

❚❤❡♦r❡♠ ■❢ ❙λ ✐s ❛ tr❡❡✲s❤✐❢t ❝♦♥tr❛❝t✐♦♥✱ t❤❡♥ ✭✐✮ ■❢ T ❤❛s ❛ r♦♦t✱ t❤❡♥ ❙λ ∈ ❈·✵(ℓ✷(❱ ))✳ ✭✐✐✮ ■❢ T ✐s r♦♦t❧❡ss✱ t❤❡♥ H✵(❙∗

λ)⊥ = ∨{❤✉ : ✉ ∈ ❱ } ✇❤❡r❡

❤✉ =

• ✈∈●❡♥(✉)

∞

• ❥=✵

λ♣❛r❥(✈) · ❡✈ ∈ ℓ✷(❱ ). ❤✉ = ❤✈ ⇐ ⇒ ✈ ∈ ●❡♥(✉) ❛♥❞ ❤✉ ❛r❡ ❡✐❣❡♥✲✈❡❝t♦rs✿ ❆∗❤✉ = ❛✉❤✉ ∀ ✉ ∈ ❱ ✇✐t❤ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❡✐❣❡♥✲✈❛❧✉❡s ❛✉ = ❤✉✷ =

• ✈∈●❡♥(✉)

∞

• ❥=✵

λ✷

♣❛r❥(✈).

❙♦✱ ❡✈❡r② ❧❡✈❡❧ ❤❛s ♦♥❡ s✉❝❤ ❤✉✳ ▼♦r❡♦✈❡r✱ ✐❢ ❤✉ ✐s ♥♦t ③❡r♦ ❢♦r ❛ ✈❡rt❡① ✉✱ t❤❡♥ ✐t ✐s ♥♦t ③❡r♦ ❢♦r ❡✈❡r② ✉ ∈ ❱ ✳

• ②ör❣② Pá❧ ●❡❤ér

❆s②♠♣t♦t✐❝ ❇❡❤❛✈✐♦✉r ❛♥❞ ❈②❝❧✐❝ Pr♦♣❡rt✐❡s ♦❢ ❚r❡❡✲s❤✐❢t ❖♣❡r❛t♦rs

❆s②♠♣t♦t✐❝ ❇❡❤❛✈✐♦✉r ❛♥❞ ❈②❝❧✐❝ Pr♦♣❡rt✐❡s ♦❢ ❚r❡❡✲s❤✐❢t ❖♣❡r❛t♦rs ❆s②♠♣t♦t✐❝ ▲✐♠✐ts ❛♥❞ ■s♦♠❡tr✐❝ ❆s②♠♣t♦t❡s ♦❢ ❚r❡❡✲s❤✐❢t ❖♣❡r❛t♦rs

❇r(T ) =

• ✉∈❱

✉ / ∈▲❡❛(T )

(|❈❤✐(✉)| − ✶) ✐s t❤❡ ❜r❛♥❝❤✐♥❣ ✐♥❞❡① ♦❢ T ✳ ❙+ ✐s t❤❡ s✐♠♣❧❡ ✉♥✐❧❛t❡r❛❧ s❤✐❢t ✭❛❧❧ ✇❡✐❣❤ts ❛r❡ ✶✮ ❛♥❞ ❙ ✐s t❤❡ s✐♠♣❧❡ ❜✐❧❛t❡r❛❧ s❤✐❢t✳ ❚❤❡② ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ❛s ♠✉❧t✐♣❧✐❝❛t✐♦♥ ♦♣❡r❛t♦rs ❜② χ(③) = ③ ♦♥ ❍✷(T) ❛♥❞ ▲✷(T)✱ r❡s♣❡❝t✐✈❡❧②✳ ❚❤❡ ✐s♦♠❡tr② ❚ ∈ B(H) ✐s ❝❛❧❧❡❞ ❝♦♠♣❧❡t❡❧② ♥♦♥✲✉♥✐t❛r② ✭❝✳♥✳✉✳ ❢♦r s❤♦rt✮ ✐❢ t❤❡ ♦♥❧② r❡❞✉❝✐♥❣ s✉❜s♣❛❝❡ ♦♥ ✇❤✐❝❤ ❚ ❛❝ts ❛s ❛ ✉♥✐t❛r② ♦♣❡r❛t♦r ✐s t❤❡ tr✐✈✐❛❧ ♦♥❡✿ {✵}✳ ❆♥ ✐s♦♠❡tr② ✐s ❝✳♥✳✉ ✐✛ ✐t ✐s ❛♥ ♦rt❤♦❣♦♥❛❧ s✉♠ ♦❢ s✐♠♣❧❡ ✉♥✐❧❛t❡r❛❧ s❤✐❢ts✳

• ②ör❣② Pá❧ ●❡❤ér

❆s②♠♣t♦t✐❝ ❇❡❤❛✈✐♦✉r ❛♥❞ ❈②❝❧✐❝ Pr♦♣❡rt✐❡s ♦❢ ❚r❡❡✲s❤✐❢t ❖♣❡r❛t♦rs

❆s②♠♣t♦t✐❝ ❇❡❤❛✈✐♦✉r ❛♥❞ ❈②❝❧✐❝ Pr♦♣❡rt✐❡s ♦❢ ❚r❡❡✲s❤✐❢t ❖♣❡r❛t♦rs ❆s②♠♣t♦t✐❝ ▲✐♠✐ts ❛♥❞ ■s♦♠❡tr✐❝ ❆s②♠♣t♦t❡s ♦❢ ❚r❡❡✲s❤✐❢t ❖♣❡r❛t♦rs

❚❤❡♦r❡♠ ❋♦r ❛ ♥♦♥✲❈✵· tr❡❡✲s❤✐❢t ❝♦♥tr❛❝t✐♦♥ ❙λ ∈ ℓ✷(❱ )✱ t❤❡ ✐s♦♠❡tr✐❝ ❛s②♠♣t♦t❡ ❯ ∈ B(ℓ✷(❱ ′)) ✐s ✉♥✐t❛r✐❧② ❡q✉✐✈❛❧❡♥t t♦✿ ✭✐✮ ❇r(T ′)+✶

❥=✶

⊕❙+✱ ✐❢ T ❤❛s ❛ r♦♦t✱ ✭✐✐✮ ❇r(T ′)

❥=✶

⊕❙+✱ ✐❢ T ❤❛s ♥♦ r♦♦t ❛♥❞ ❯ ✐s ❛ ❝✳♥✳✉✳ ✐s♦♠❡tr②✱ ✐✳❡✳✿

• ✈′∈●❡♥T ′(✉′)

∞

❥=✵ β✷ ♣❛r❥(✈′) = ✵ ❢♦r s♦♠❡ ✭❛♥❞ t❤❡♥ ❢♦r ❡✈❡r②✮

✉′ ∈ ❱ ′✱ ✭✐✐✐✮ ❙ ⊕ ❇r(T ′)

❥=✶

⊕❙+✱ ✐❢ T ❤❛s ♥♦ r♦♦t ❛♥❞ ❯ ✐s ♥♦t ❛ ❝✳♥✳✉✳ ✐s♦♠❡tr②✳

• ②ör❣② Pá❧ ●❡❤ér

❆s②♠♣t♦t✐❝ ❇❡❤❛✈✐♦✉r ❛♥❞ ❈②❝❧✐❝ Pr♦♣❡rt✐❡s ♦❢ ❚r❡❡✲s❤✐❢t ❖♣❡r❛t♦rs

❆s②♠♣t♦t✐❝ ❇❡❤❛✈✐♦✉r ❛♥❞ ❈②❝❧✐❝ Pr♦♣❡rt✐❡s ♦❢ ❚r❡❡✲s❤✐❢t ❖♣❡r❛t♦rs ❆s②♠♣t♦t✐❝ ▲✐♠✐ts ❛♥❞ ■s♦♠❡tr✐❝ ❆s②♠♣t♦t❡s ♦❢ ❚r❡❡✲s❤✐❢t ❖♣❡r❛t♦rs

❚❤❡♦r❡♠ ❙✉♣♣♦s❡ t❤❛t t❤❡ ❝♦♥tr❛❝t✐✈❡ tr❡❡✲s❤✐❢t ♦♣❡r❛t♦r ❙λ ✐s ♥♦t ✐♥ t❤❡ ❝❧❛ss ❈·✵✳ ❚❤❡♥ T ❤❛s ♥♦ r♦♦t ❛♥❞ t❤❡ ✐s♦♠❡tr② ❯∗ ❛❝ts ❛s ❢♦❧❧♦✇s✿ ❯∗❤✉ = √❛✉ √❛♣❛r(✉) · ❤♣❛r(✉), ✇❤❡r❡ ❤✉ = ✵ ❢♦r ❡✈❡r② ✉ ∈ ❱ ✳ ■♥ ❢❛❝t ❯∗ ✐s ❛ s✐♠♣❧❡ ✉♥✐❧❛t❡r❛❧ s❤✐❢t✱ ✐❢ ❈❤✐(●❡♥(✉)) = ∅ ❢♦r s♦♠❡ ✉ ∈ ❱ ✱ ❛♥❞ ❛ s✐♠♣❧❡ ❜✐❧❛t❡r❛❧ s❤✐❢t ❡❧s❡✇❤❡r❡✳

• ②ör❣② Pá❧ ●❡❤ér

❆s②♠♣t♦t✐❝ ❇❡❤❛✈✐♦✉r ❛♥❞ ❈②❝❧✐❝ Pr♦♣❡rt✐❡s ♦❢ ❚r❡❡✲s❤✐❢t ❖♣❡r❛t♦rs

❆s②♠♣t♦t✐❝ ❇❡❤❛✈✐♦✉r ❛♥❞ ❈②❝❧✐❝ Pr♦♣❡rt✐❡s ♦❢ ❚r❡❡✲s❤✐❢t ❖♣❡r❛t♦rs ❆s②♠♣t♦t✐❝ ▲✐♠✐ts ❛♥❞ ■s♦♠❡tr✐❝ ❆s②♠♣t♦t❡s ♦❢ ❚r❡❡✲s❤✐❢t ❖♣❡r❛t♦rs

❈♦r♦❧❧❛r② ❈♦♥s✐❞❡r t❤❡ tr❡❡✲s❤✐❢t ❝♦♥tr❛❝t✐♦♥ ❙λ✳ ❚❤❡♥ t❤❡ ❢♦❧❧♦✇✐♥❣s ❤♦❧❞ ✭✐✮ ❙λ ✐s s✐♠✐❧❛r t♦ ❛♥ ✐s♦♠❡tr② ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐♥❢{α✉ : ✉ ∈ ❱ } > ✵✳ ✭✐✐✮ ❙λ ✐s s✐♠✐❧❛r t♦ ❛ ❝♦✲✐s♦♠❡tr② ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐t ✐s ❛ ❜✐❧❛t❡r❛❧ ✇❡✐❣❤t❡❞ s❤✐❢t ✇✐t❤ ∞

❥=−∞ λ❥ > ✵✱ ♦r ✐t ✐s ❛ ✉♥✐❧❛t❡r❛❧

✇❡✐❣❤t❡❞ ❜❛❝❦✇❛r❞ s❤✐❢t ✇✐t❤ ∞

❥=✵ λ❥ > ✵✳ ❚❤❡♥ ✐t ✐s s✐♠✐❧❛r

t♦ t❤❡ s✐♠♣❧❡ ❜✐❧❛t❡r❛❧ ♦r t❤❡ s✐♠♣❧❡ ❜❛❝❦✇❛r❞ s❤✐❢t ♦♣❡r❛t♦r✱ r❡s♣❡❝t✐✈❡❧②✳

• ②ör❣② Pá❧ ●❡❤ér

❆s②♠♣t♦t✐❝ ❇❡❤❛✈✐♦✉r ❛♥❞ ❈②❝❧✐❝ Pr♦♣❡rt✐❡s ♦❢ ❚r❡❡✲s❤✐❢t ❖♣❡r❛t♦rs

❆s②♠♣t♦t✐❝ ❇❡❤❛✈✐♦✉r ❛♥❞ ❈②❝❧✐❝ Pr♦♣❡rt✐❡s ♦❢ ❚r❡❡✲s❤✐❢t ❖♣❡r❛t♦rs ❈②❝❧✐❝ Pr♦♣❡rt✐❡s ❈②❝❧✐❝✐t② ♦❢ ❚r❡❡✲s❤✐❢t ❖♣❡r❛t♦rs

❊❛s② t♦ s❡❡✿ ■❢ ❝♦ − ❞✐♠(R(❚)−) > ✶ = ⇒ ❚ ❤❛s ♥♦ ❝②❝❧✐❝ ✈❡❝t♦rs✳ ⇓ ✐❢ T ❤❛s ❛ r♦♦t ❛♥❞ ❇r(T ) > ✵ = ⇒ ❙λ ❤❛s ♥♦ ❝②❝❧✐❝ ✈❡❝t♦rs✱ ✐❢ T ✐s r♦♦t❧❡ss ❛♥❞ ❇r(T ) > ✶ = ⇒ ❙λ ❤❛s ♥♦ ❝②❝❧✐❝ ✈❡❝t♦rs✳ ❚❤❡ ♣✉r❡ tr❡❡✲s❤✐❢t ❝❛s❡ ✐s ✇❤❡♥ ❇r(T ) = ✶ ❛♥❞ T ❤❛s ♥♦ r♦♦t✳

• ②ör❣② Pá❧ ●❡❤ér

❆s②♠♣t♦t✐❝ ❇❡❤❛✈✐♦✉r ❛♥❞ ❈②❝❧✐❝ Pr♦♣❡rt✐❡s ♦❢ ❚r❡❡✲s❤✐❢t ❖♣❡r❛t♦rs

❆s②♠♣t♦t✐❝ ❇❡❤❛✈✐♦✉r ❛♥❞ ❈②❝❧✐❝ Pr♦♣❡rt✐❡s ♦❢ ❚r❡❡✲s❤✐❢t ❖♣❡r❛t♦rs ❈②❝❧✐❝ Pr♦♣❡rt✐❡s ❈②❝❧✐❝✐t② ♦❢ ❚r❡❡✲s❤✐❢t ❖♣❡r❛t♦rs

❇r(T ) = ✵ = ⇒ ❙λ ✐s ❛ ✇❡✐❣❤t❡❞ ❜✐❧❛t❡r❛❧ s❤✐❢t ✭♥♦ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ❢♦r ❝②❝❧✐❝✐t② ✐s ❦♥♦✇♥✮ ✭❢♦r s✉♣❡r❝②❝❧✐❝✐t②✱ ❤②♣❡r❝②❝❧✐❝✐t② ✳ ✳ ✳ t❤❡r❡ ❛r❡ ❝❤❛r❛❝t❡r✐③❛t✐♦♥s✮✱ ❛ ✇❡✐❣❤t❡❞ ✉♥✐❧❛t❡r❛❧ s❤✐❢t ✭❡❛s②✿ ❛❧✇❛②s ❝②❝❧✐❝✮✱ ❛ ✇❡✐❣❤t❡❞ ❜❛❝❦✇❛r❞ s❤✐❢t ✭❛❧✇❛②s ❝②❝❧✐❝✱ ❧❛t❡r✮✱ ❛ ❝②❝❧✐❝ ♥✐❧♣♦t❡♥t ♦♣❡r❛t♦r ❛❝t✐♥❣ ♦♥ ❛ ✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧ s♣❛❝❡✳ ❚❤❡r❡ ❛r❡ ❝②❝❧✐❝ ❛♥❞ ♥♦♥✲❝②❝❧✐❝ ❜✐❧❛t❡r❛❧ s❤✐❢ts✱ s❡❡✿ ❆✳ ▲✳ ❙❤✐❡❧❞s✱ ❲❡✐❣❤t❡❞ s❤✐❢t ♦♣❡r❛t♦rs ❛♥❞ ❛♥❛❧②t✐❝ ❢✉♥❝t✐♦♥ t❤❡♦r②✱ ❚♦♣✐❝s ✐♥ ❖♣❡r❛t♦r ❚❤❡♦r②✱ ▼❛t❤✳ ❙✉r✈❡②s ✶✸✱ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳✱ Pr♦✈✐❞❡♥❝❡✱ ❘✳ ■✳✱ ✶✾✼✹✱ ✹✾✕✶✷✽✳

• ②ör❣② Pá❧ ●❡❤ér

❆s②♠♣t♦t✐❝ ❇❡❤❛✈✐♦✉r ❛♥❞ ❈②❝❧✐❝ Pr♦♣❡rt✐❡s ♦❢ ❚r❡❡✲s❤✐❢t ❖♣❡r❛t♦rs

❆s②♠♣t♦t✐❝ ❇❡❤❛✈✐♦✉r ❛♥❞ ❈②❝❧✐❝ Pr♦♣❡rt✐❡s ♦❢ ❚r❡❡✲s❤✐❢t ❖♣❡r❛t♦rs ❈②❝❧✐❝ Pr♦♣❡rt✐❡s ❈②❝❧✐❝✐t② ♦❢ ❚r❡❡✲s❤✐❢t ❖♣❡r❛t♦rs

❚❤❡♦r❡♠ ❙✉♣♣♦s❡ t❤❛t ❇ ✐s ❛ ✇❡✐❣❤t❡❞ ❜❛❝❦✇❛r❞ s❤✐❢t ♦❢ ❝♦✉♥t❛❜❧❡ ♠✉❧t✐♣❧✐❝✐t②✳ ❚❤❡♥ ❇ ✐s ❝②❝❧✐❝ ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡r❡ ✐s ❛t ♠♦st ♦♥❡ ③❡r♦ ✇❡✐❣❤t✳ ■♥ ❝❛s❡ t❤❡ ♠✉❧t✐♣❧✐❝✐t② ✐s ✶ t❤❡ ✇❛s ♦❜t❛✐♥❡❞ ❜② ❩✳ ●✉❛♥❣ ❍✉❛ ✐♥ ✶✾✽✹ ✭✇r✐tt❡♥ ✐♥ ❈❤✐♥❡s❡✮✳

• ②ör❣② Pá❧ ●❡❤ér

❆s②♠♣t♦t✐❝ ❇❡❤❛✈✐♦✉r ❛♥❞ ❈②❝❧✐❝ Pr♦♣❡rt✐❡s ♦❢ ❚r❡❡✲s❤✐❢t ❖♣❡r❛t♦rs

❆s②♠♣t♦t✐❝ ❇❡❤❛✈✐♦✉r ❛♥❞ ❈②❝❧✐❝ Pr♦♣❡rt✐❡s ♦❢ ❚r❡❡✲s❤✐❢t ❖♣❡r❛t♦rs ❈②❝❧✐❝ Pr♦♣❡rt✐❡s ❈②❝❧✐❝✐t② ♦❢ ❚r❡❡✲s❤✐❢t ❖♣❡r❛t♦rs

❚❤❡♦r❡♠ ■❢ t❤❡ ❞✐r❡❝t❡❞ tr❡❡ T ❤❛s ♥♦ r♦♦t✱ ❇r(T ) = ✶ ❛♥❞ ❤❛✈❡ ✷ ❧❡❛✈❡s✱ t❤❡♥ ❡✈❡r② ❜♦✉♥❞❡❞ tr❡❡✲s❤✐❢t ♦♣❡r❛t♦r ♦♥ ✐t ✐s ❝②❝❧✐❝✳ Pr♦♦❢✳ ❙λ ✐s s✐♠✐❧❛r t♦ ❛ ❜❛❝❦✇❛r❞ s❤✐❢t✱ ✇❤✐❝❤ ❤❛s ❛t ♠♦st ♦♥❡ ③❡r♦ ✇❡✐❣❤t✳

• ②ör❣② Pá❧ ●❡❤ér

❆s②♠♣t♦t✐❝ ❇❡❤❛✈✐♦✉r ❛♥❞ ❈②❝❧✐❝ Pr♦♣❡rt✐❡s ♦❢ ❚r❡❡✲s❤✐❢t ❖♣❡r❛t♦rs

❆s②♠♣t♦t✐❝ ❇❡❤❛✈✐♦✉r ❛♥❞ ❈②❝❧✐❝ Pr♦♣❡rt✐❡s ♦❢ ❚r❡❡✲s❤✐❢t ❖♣❡r❛t♦rs ❈②❝❧✐❝ Pr♦♣❡rt✐❡s ❈②❝❧✐❝✐t② ♦❢ ❚r❡❡✲s❤✐❢t ❖♣❡r❛t♦rs

❚❤❡♦r❡♠ ❙✉♣♣♦s❡ T ❤❛s ❛ ✉♥✐q✉❡ ❧❡❛❢✳ ❆ tr❡❡✲s❤✐❢t ♦♣❡r❛t♦r ❙λ ♦♥ T ✐s ❝②❝❧✐❝ ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡ ❜✐❧❛t❡r❛❧ s❤✐❢t ❲ ✇✐t❤ ✇❡✐❣❤ts {λ♥}∞

♥=−∞ ✐s

❝②❝❧✐❝✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ✐❢ ❙λ / ∈ ❈·✵(ℓ✷(❱ ))✱ t❤❡♥ ❙λ ✐s ❝②❝❧✐❝✳

• ②ör❣② Pá❧ ●❡❤ér

❆s②♠♣t♦t✐❝ ❇❡❤❛✈✐♦✉r ❛♥❞ ❈②❝❧✐❝ Pr♦♣❡rt✐❡s ♦❢ ❚r❡❡✲s❤✐❢t ❖♣❡r❛t♦rs

❆s②♠♣t♦t✐❝ ❇❡❤❛✈✐♦✉r ❛♥❞ ❈②❝❧✐❝ Pr♦♣❡rt✐❡s ♦❢ ❚r❡❡✲s❤✐❢t ❖♣❡r❛t♦rs ❈②❝❧✐❝ Pr♦♣❡rt✐❡s ❈②❝❧✐❝✐t② ♦❢ ❚r❡❡✲s❤✐❢t ❖♣❡r❛t♦rs

Pr♦♣♦s✐t✐♦♥ ❚❤❡ ♦♣❡r❛t♦r ❙ ⊕ ❙+ ❤❛s ♥♦ ❝②❝❧✐❝ ✈❡❝t♦rs✳ ❚❤❡♦r❡♠ ❙✉♣♣♦s❡ t❤❛t T ✐s r♦♦t❧❡ss ❛♥❞ ❇r(T ) = ✶✳ ■❢ t❤❡ tr❡❡✲s❤✐❢t ❝♦♥tr❛❝t✐♦♥ ❙λ ✐s ♦❢ ❝❧❛ss ❈✶·✱ t❤❡♥ ✐t ❤❛s ♥♦ ❝②❝❧✐❝ ✈❡❝t♦rs✳ Pr♦♦❢✳ ❚❤❡ ✐s♦♠❡tr✐❝ ❛s②♠♣t♦t❡ ♦❢ ❙λ ✐s ✉♥✐t❛r✐❧② ❡q✉✐✈❛❧❡♥t t♦ ❙ ⊕ ❙+✳ ❖♥ t❤✐s T t❤❡ ❝♦♥tr❛r② ♠❛② ❛❧s♦ ❤❛♣♣❡♥✳

• ②ör❣② Pá❧ ●❡❤ér

❆s②♠♣t♦t✐❝ ❇❡❤❛✈✐♦✉r ❛♥❞ ❈②❝❧✐❝ Pr♦♣❡rt✐❡s ♦❢ ❚r❡❡✲s❤✐❢t ❖♣❡r❛t♦rs

❆s②♠♣t♦t✐❝ ❇❡❤❛✈✐♦✉r ❛♥❞ ❈②❝❧✐❝ Pr♦♣❡rt✐❡s ♦❢ ❚r❡❡✲s❤✐❢t ❖♣❡r❛t♦rs ❈②❝❧✐❝ Pr♦♣❡rt✐❡s ❙✐♠✐❧❛r✐t② t♦ ♦rt❤♦❣♦♥❛❧ s✉♠ ♦❢ ❜✐✲ ❛♥❞ ✉♥✐❧❛t❡r❛❧ s❤✐❢ts

❚❤❡ ❣r❛♣❤s ♦❢ ❙λ ❛♥❞ ❲

❣❦ = ❦

❥=✶ ✶ λ❥ · ❡❦ − ❦ ❥=✶ ✶ λ❥′ · ❡❦′

(❦ ∈ N) ✳

• ②ör❣② Pá❧ ●❡❤ér

❆s②♠♣t♦t✐❝ ❇❡❤❛✈✐♦✉r ❛♥❞ ❈②❝❧✐❝ Pr♦♣❡rt✐❡s ♦❢ ❚r❡❡✲s❤✐❢t ❖♣❡r❛t♦rs

❆s②♠♣t♦t✐❝ ❇❡❤❛✈✐♦✉r ❛♥❞ ❈②❝❧✐❝ Pr♦♣❡rt✐❡s ♦❢ ❚r❡❡✲s❤✐❢t ❖♣❡r❛t♦rs ❈②❝❧✐❝ Pr♦♣❡rt✐❡s ❙✐♠✐❧❛r✐t② t♦ ♦rt❤♦❣♦♥❛❧ s✉♠ ♦❢ ❜✐✲ ❛♥❞ ✉♥✐❧❛t❡r❛❧ s❤✐❢ts

❚❤❡♦r❡♠ ■❢ ❦

❥=✶ λ❥′ λ❥ : ❦ ∈ N

• ✐s ❜♦✉♥❞❡❞✱ t❤❡♥ ❙λ ✐s s✐♠✐❧❛r t♦ ❲ ✳

❈♦r♦❧❧❛r② ■❢ ❙λ / ∈ ❈✵·(ℓ✷(❱ ))✱ t❤❡♥ ✐t ✐s s✐♠✐❧❛r t♦ ❲ ✳ ❚❤❡♦r❡♠ ❚❤❡r❡ ✐s ❛ tr❡❡✲s❤✐❢t ♦♣❡r❛t♦r ♦♥ t❤❡ ♣r❡✈✐♦✉s ❞✐r❡❝t❡❞ tr❡❡ ✇❤✐❝❤ ✐s ❝②❝❧✐❝✳

• ②ör❣② Pá❧ ●❡❤ér

❆s②♠♣t♦t✐❝ ❇❡❤❛✈✐♦✉r ❛♥❞ ❈②❝❧✐❝ Pr♦♣❡rt✐❡s ♦❢ ❚r❡❡✲s❤✐❢t ❖♣❡r❛t♦rs

❆s②♠♣t♦t✐❝ ❇❡❤❛✈✐♦✉r ❛♥❞ ❈②❝❧✐❝ Pr♦♣❡rt✐❡s ♦❢ ❚r❡❡✲s❤✐❢t ❖♣❡r❛t♦rs ❈②❝❧✐❝ Pr♦♣❡rt✐❡s ❈②❝❧✐❝✐t② ♦❢ t❤❡ ❆❞❥♦✐♥t

❙+

❦ := ❙+ ⊕ · · · ⊕ ❙+

• ❦ t✐♠❡s

✭❦ ∈ N✮✱ ❙+

ℵ✵ := ❙+ ⊕ ❙+ ⊕ . . .

• ℵ✵ ♠❛♥②

✳ ❚❤❡♦r❡♠ ❚❤❡ ♦♣❡r❛t♦r ❙ ⊕ (❙+

❦ )∗ ✐s ❝②❝❧✐❝ ❢♦r ❡✈❡r② ❦ ∈ N✳

❚❤❡♦r❡♠ ❚❤❡ ❢♦❧❧♦✇✐♥❣s ❛r❡ ✈❛❧✐❞✿ ✭✐✮ ■❢ T ❤❛s ❛ r♦♦t ❛♥❞ t❤❡ tr❡❡✲s❤✐❢t ❝♦♥tr❛❝t✐♦♥ ❙λ ♦♥ ✐t ✐s ♦❢ ❝❧❛ss ❈✶·✱ t❤❡♥ ❙∗

λ ✐s ❝②❝❧✐❝✳

✭✐✐✮ ■❢ T ✐s r♦♦t❧❡ss✱ ❇r(T ) < ∞ ❛♥❞ t❤❡ tr❡❡✲s❤✐❢t ❝♦♥tr❛❝t✐♦♥ ❙λ ♦♥ ✐t ✐s ♦❢ ❝❧❛ss ❈✶·✱ t❤❡♥ ❙∗

λ ✐s ❝②❝❧✐❝✳

• ②ör❣② Pá❧ ●❡❤ér

❆s②♠♣t♦t✐❝ ❇❡❤❛✈✐♦✉r ❛♥❞ ❈②❝❧✐❝ Pr♦♣❡rt✐❡s ♦❢ ❚r❡❡✲s❤✐❢t ❖♣❡r❛t♦rs

❆s②♠♣t♦t✐❝ ❇❡❤❛✈✐♦✉r ❛♥❞ ❈②❝❧✐❝ Pr♦♣❡rt✐❡s ♦❢ ❚r❡❡✲s❤✐❢t ❖♣❡r❛t♦rs ❈②❝❧✐❝ Pr♦♣❡rt✐❡s ❈②❝❧✐❝✐t② ♦❢ t❤❡ ❆❞❥♦✐♥t

◗✉❡st✐♦♥ ■s t❤❡ ♦♣❡r❛t♦r ❙ ⊕ (❙+

ℵ✵)∗ ❝②❝❧✐❝❄

• ②ör❣② Pá❧ ●❡❤ér

❆s②♠♣t♦t✐❝ ❇❡❤❛✈✐♦✉r ❛♥❞ ❈②❝❧✐❝ Pr♦♣❡rt✐❡s ♦❢ ❚r❡❡✲s❤✐❢t ❖♣❡r❛t♦rs

❚❤✐s r❡s❡❛r❝❤ ✇❛s r❡❛❧✐③❡❞ ✐♥ t❤❡ ❢r❛♠❡s ♦❢ ❚➪▼❖P ✹✳✷✳✹✳ ❆✴✷✲✶✶✲✶✲✷✵✶✷✲✵✵✵✶ ✑◆❛t✐♦♥❛❧ ❊①❝❡❧❧❡♥❝❡ Pr♦❣r❛♠ ✲ ❊❧❛❜♦r❛t✐♥❣ ❛♥❞ ♦♣❡r❛t✐♥❣ ❛♥ ✐♥❧❛♥❞ st✉❞❡♥t ❛♥❞ r❡s❡❛r❝❤❡r ♣❡rs♦♥❛❧ s✉♣♣♦rt s②st❡♠✑✳ ❚❤❡ ♣r♦❥❡❝t ✇❛s s✉❜s✐❞✐③❡❞ ❜② t❤❡ ❊✉r♦♣❡❛♥ ❯♥✐♦♥ ❛♥❞ ❝♦✲✜♥❛♥❝❡❞ ❜② t❤❡ ❊✉r♦♣❡❛♥ ❙♦❝✐❛❧ ❋✉♥❞

❚❤❛♥❦ ❨♦✉ ❋♦r ❨♦✉r ❑✐♥❞ ❆tt❡♥t✐♦♥✦