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❯♥✐t❛r② s✐♠✐❧❛r✐t②✳ ◆❡✇ ❝r✐t❡r✐❛ ❚❛t✐❛♥❛ ●✳ ●❡r❛s✐♠♦✈❛ ❑✐❡✈ ◆❛t✐♦♥❛❧ ❚❛r❛s ❙❤❡✈❝❤❡♥❦♦ ❯♥✐✈❡rs✐t②✱ ❯❦r❛✐♥❡ ❇❡♦❣r❛❞✱ ✷✵✶✷ ❚❛t✐❛♥❛ ●✳ ●❡r❛s✐♠♦✈❛ ❯♥✐t❛r② s✐♠✐❧❛r✐t②✳ ◆❡✇ ❝r✐t❡r✐❛

❘❡❢❡r❡♥❝❡s ❬❋●❙✬✶✶❪ ❉✳ ❋❛r❡♥✐❝❦✱ ❚✳ ●✳ ●❡r❛s✐♠♦✈❛✱ ◆✳ ❙❤✈❛✐ ✱ ❆ ❝♦♠♣❧❡t❡ ✉♥✐t❛r② s✐♠✐❧❛r✐t② ✐♥✈❛r✐❛♥t ❢♦r ✉♥✐❝❡❧❧✉❧❛r ♠❛tr✐❝❡s✱ ▲✐♥❡❛r ❆❧❣❡❜r❛ ❆♣♣❧✳✱ ✹✸✺ ✭✷✵✶✶✮ ✹✵✾✲✹✶✾ ✳ ❬❋❋●❙❙✬✶✶❪ ❉✳ ❋❛r❡♥✐❝❦✱ ❱✳ ❋✉t♦r♥②✱ ❚✳ ●✳ ●❡r❛s✐♠♦✈❛✱ ❱✳ ❱✳ ❙❡r❣❡✐❝❤✉❦✱ ◆✳ ❙❤✈❛✐ ✱ ❈r✐t❡r✐♦♥ ♦❢ ✉♥✐t❛r② s✐♠✐❧❛r✐t② ❢♦r ✉♣♣❡r tr✐❛♥❣✉❧❛r ♠❛tr✐❝❡s ✐♥ ❣❡♥❡r❛❧ ♣♦s✐t✐♦♥✳ ▲✐♥❡❛r ❆❧❣❡❜r❛ ❆♣♣❧✳✱ ✹✸✺ ✭✷✵✶✶✮ ✶✸✺✻✲✶✸✻✾ ✳ ❬●✬✶✷❪ ❚✳●✳ ●❡r❛s✐♠♦✈❛ ✱ ❯♥✐t❛r② s✐♠✐❧❛r✐t② t♦ ❛ ♥♦r♠❛❧ ♠❛tr✐①✳ ▲✐♥❡❛r ❆❧❣❡❜r❛ ❆♣♣❧✳✱ ✹✸✻ ✭✷✵✶✷✮ ✸✼✼✼✲✸✼✽✸ ✳ ❚❛t✐❛♥❛ ●✳ ●❡r❛s✐♠♦✈❛ ❯♥✐t❛r② s✐♠✐❧❛r✐t②✳ ◆❡✇ ❝r✐t❡r✐❛

■♥✈❛r✐❛♥ts ✉♥❞❡r ✉♥✐t❛r② s✐♠✐❧❛r✐t②✿ ♥♦r♠ tr❛❝❡ s♣❡❝tr✉♠ ♥✉♠❡r✐❝❛❧ r❛♥❣❡ Pr♦❜❧❡♠ ❞❡✜♥✐t✐♦♥ ❆ ❝❧❛ss✐❝❛❧ ♣r♦❜❧❡♠ ♦❢ ♦♣❡r❛t♦r t❤❡♦r②✿ ■❢ R ❛♥❞ S ❛r❡ ♦♣❡r❛t♦rs ❛❝t✐♥❣ ♦♥ ❛ ❝♦♠♣❧❡① ❍✐❧❜❡rt s♣❛❝❡ H ✱ t❤❡♥ ❤♦✇ ❝❛♥ ♦♥❡ ❞❡t❡r♠✐♥❡ ✇❤❡t❤❡r R ❛♥❞ S ❛r❡ ✉♥✐t❛r✐❧② s✐♠✐❧❛r❄ ❚❛t✐❛♥❛ ●✳ ●❡r❛s✐♠♦✈❛ ❯♥✐t❛r② s✐♠✐❧❛r✐t②✳ ◆❡✇ ❝r✐t❡r✐❛

■♥✈❛r✐❛♥ts ✉♥❞❡r ✉♥✐t❛r② s✐♠✐❧❛r✐t②✿ ♥♦r♠ tr❛❝❡ s♣❡❝tr✉♠ ♥✉♠❡r✐❝❛❧ r❛♥❣❡ Pr♦❜❧❡♠ ❞❡✜♥✐t✐♦♥ ❆ ❝❧❛ss✐❝❛❧ ♣r♦❜❧❡♠ ♦❢ ♦♣❡r❛t♦r t❤❡♦r②✿ ■❢ R ❛♥❞ S ❛r❡ ♦♣❡r❛t♦rs ❛❝t✐♥❣ ♦♥ ❛ ❝♦♠♣❧❡① ❍✐❧❜❡rt s♣❛❝❡ H ✱ t❤❡♥ ❤♦✇ ❝❛♥ ♦♥❡ ❞❡t❡r♠✐♥❡ ✇❤❡t❤❡r R ❛♥❞ S ❛r❡ ✉♥✐t❛r✐❧② s✐♠✐❧❛r❄ ❖♣❡r❛t♦rs R ❛♥❞ S ❛r❡ ❝❛❧❧❡❞ ✉♥✐t❛r② s✐♠✐❧❛r✱ ✐❢ t❤❡r❡ ✐s ❛ ✉♥✐t❛r② ♦♣❡r❛t♦r U s✉❝❤ t❤❛t S = U ∗ RU ✳ ❚❛t✐❛♥❛ ●✳ ●❡r❛s✐♠♦✈❛ ❯♥✐t❛r② s✐♠✐❧❛r✐t②✳ ◆❡✇ ❝r✐t❡r✐❛

■♥✈❛r✐❛♥ts ✉♥❞❡r ✉♥✐t❛r② s✐♠✐❧❛r✐t②✿ ♥♦r♠ tr❛❝❡ s♣❡❝tr✉♠ ♥✉♠❡r✐❝❛❧ r❛♥❣❡ Pr♦❜❧❡♠ ❞❡✜♥✐t✐♦♥ ❆ ❝❧❛ss✐❝❛❧ ♣r♦❜❧❡♠ ♦❢ ♦♣❡r❛t♦r t❤❡♦r②✿ ■❢ R ❛♥❞ S ❛r❡ ♦♣❡r❛t♦rs ❛❝t✐♥❣ ♦♥ ❛ ❝♦♠♣❧❡① ❍✐❧❜❡rt s♣❛❝❡ H ✱ t❤❡♥ ❤♦✇ ❝❛♥ ♦♥❡ ❞❡t❡r♠✐♥❡ ✇❤❡t❤❡r R ❛♥❞ S ❛r❡ ✉♥✐t❛r✐❧② s✐♠✐❧❛r❄ ▼♦r❡ ♣r❡❝✐s❡❧②✱ t❤❡ ♣r♦❜❧❡♠ ✐s t♦ ✜♥❞ ❛ s❡t ♦❢ ✐♥✈❛r✐❛♥ts t❤❛t ❝♦♠♣❧❡t❡❧② ❞❡t❡r♠✐♥❡ ❛♥ ♦♣❡r❛t♦r ✉♣ t♦ ✉♥✐t❛r② s✐♠✐❧❛r✐t②✳ ❚❛t✐❛♥❛ ●✳ ●❡r❛s✐♠♦✈❛ ❯♥✐t❛r② s✐♠✐❧❛r✐t②✳ ◆❡✇ ❝r✐t❡r✐❛

Pr♦❜❧❡♠ ❞❡✜♥✐t✐♦♥ ❆ ❝❧❛ss✐❝❛❧ ♣r♦❜❧❡♠ ♦❢ ♦♣❡r❛t♦r t❤❡♦r②✿ ■❢ R ❛♥❞ S ❛r❡ ♦♣❡r❛t♦rs ❛❝t✐♥❣ ♦♥ ❛ ❝♦♠♣❧❡① ❍✐❧❜❡rt s♣❛❝❡ H ✱ t❤❡♥ ❤♦✇ ❝❛♥ ♦♥❡ ❞❡t❡r♠✐♥❡ ✇❤❡t❤❡r R ❛♥❞ S ❛r❡ ✉♥✐t❛r✐❧② s✐♠✐❧❛r❄ ▼♦r❡ ♣r❡❝✐s❡❧②✱ t❤❡ ♣r♦❜❧❡♠ ✐s t♦ ✜♥❞ ❛ s❡t ♦❢ ✐♥✈❛r✐❛♥ts t❤❛t ❝♦♠♣❧❡t❡❧② ❞❡t❡r♠✐♥❡ ❛♥ ♦♣❡r❛t♦r ✉♣ t♦ ✉♥✐t❛r② s✐♠✐❧❛r✐t②✳ ■♥✈❛r✐❛♥ts ✉♥❞❡r ✉♥✐t❛r② s✐♠✐❧❛r✐t②✿ ♥♦r♠ tr❛❝❡ s♣❡❝tr✉♠ ♥✉♠❡r✐❝❛❧ r❛♥❣❡ ❚❛t✐❛♥❛ ●✳ ●❡r❛s✐♠♦✈❛ ❯♥✐t❛r② s✐♠✐❧❛r✐t②✳ ◆❡✇ ❝r✐t❡r✐❛

❋✐♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ ❍✐❧❜❡rt s♣❛❝❡ ❙♣❡❝❤t✬s t❤❡♦r❡♠ ✭✶✾✹✵✮ ❖♣❡r❛t♦rs R ❛♥❞ S ♦♥ ❛ ✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ ❍✐❧❜❡rt s♣❛❝❡ H ❛r❡ ✉♥✐t❛r✐❧② s✐♠✐❧❛r ✐❢ ❛♥❞ ♦♥❧② ✐❢ trace ω ( R, R ∗ ) = trace ω ( S, S ∗ ) ❢♦r ❡✈❡r② ✇♦r❞ ω ( s, t ) ✐♥ t✇♦ ♥♦♥❝♦♠♠✉t✐♥❣ ✈❛r✐❛❜❧❡s✳ ❚❛t✐❛♥❛ ●✳ ●❡r❛s✐♠♦✈❛ ❯♥✐t❛r② s✐♠✐❧❛r✐t②✳ ◆❡✇ ❝r✐t❡r✐❛

❋✐♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ ❍✐❧❜❡rt s♣❛❝❡ ❙♣❡❝❤t✬s t❤❡♦r❡♠ ✭✶✾✹✵✮ ❖♣❡r❛t♦rs R ❛♥❞ S ♦♥ ❛ ✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ ❍✐❧❜❡rt s♣❛❝❡ H ❛r❡ ✉♥✐t❛r✐❧② s✐♠✐❧❛r ✐❢ ❛♥❞ ♦♥❧② ✐❢ trace ω ( R, R ∗ ) = trace ω ( S, S ∗ ) ❢♦r ❡✈❡r② ✇♦r❞ ω ( s, t ) ✐♥ t✇♦ ♥♦♥❝♦♠♠✉t✐♥❣ ✈❛r✐❛❜❧❡s✳ P❛♣♣❛❝❡♥❛✬s r❡str✐❝t✐♦♥ ✭✶✾✾✼✮ ❖♣❡r❛t♦rs R ❛♥❞ S ♦♥ ❛♥ n ✲❞✐♠❡♥s✐♦♥❛❧ ❍✐❧❜❡rt s♣❛❝❡ H ❛r❡ ✉♥✐t❛r✐❧② s✐♠✐❧❛r ✐❢ ❛♥❞ ♦♥❧② ✐❢ trace ω ( R, R ∗ ) = trace ω ( S, S ∗ ) ❢♦r ❡✈❡r② ✇♦r❞ ω ( s, t ) ✐♥ t✇♦ ♥♦♥❝♦♠♠✉t✐♥❣ ✈❛r✐❛❜❧❡s ✇❤♦s❡ ❧❡♥❣t❤ ✐s ❛t ♠♦st � 2 n 2 n − 1 + 1 4 + n n 2 − 2 ❚❛t✐❛♥❛ ●✳ ●❡r❛s✐♠♦✈❛ ❯♥✐t❛r② s✐♠✐❧❛r✐t②✳ ◆❡✇ ❝r✐t❡r✐❛

❋✐♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ ❍✐❧❜❡rt s♣❛❝❡ ❙♣❡❝❤t✬s t❤❡♦r❡♠ ✭✶✾✹✵✮ ❖♣❡r❛t♦rs R ❛♥❞ S ♦♥ ❛ ✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ ❍✐❧❜❡rt s♣❛❝❡ H ❛r❡ ✉♥✐t❛r✐❧② s✐♠✐❧❛r ✐❢ ❛♥❞ ♦♥❧② ✐❢ trace ω ( R, R ∗ ) = trace ω ( S, S ∗ ) ❢♦r ❡✈❡r② ✇♦r❞ ω ( s, t ) ✐♥ t✇♦ ♥♦♥❝♦♠♠✉t✐♥❣ ✈❛r✐❛❜❧❡s✳ ❚❤✐s t❤❡♦r❡♠ ✐s ❡①t❡♥❞❡❞ t♦ ❝♦♠♣❛❝t ♦♣❡r❛t♦rs ✐♥ tr❛❝❡✲ ❛♥❞ ❙❝❤♠✐❞t ❝❧❛ss❡s ♦♥ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ ❍✐❧❜❡rt s♣❛❝❡s ❛♥❞ ❝❛♥♥♦t ❜❡ ❡①t❡♥❞❡❞ t♦ ❛❧❧ ❝♦♠♣❛❝t ♦♣❡r❛t♦rs✳ ❚❛t✐❛♥❛ ●✳ ●❡r❛s✐♠♦✈❛ ❯♥✐t❛r② s✐♠✐❧❛r✐t②✳ ◆❡✇ ❝r✐t❡r✐❛

❆r✈❡s♦♥✬s ❝r✐t❡r✐♦♥ ❋♦r ❡❛❝❤ ♠❛tr✐① ♣♦❧②♥♦♠✐❛❧ p ( x ) = A 0 + A 1 x + · · · + A t x t ∈ C k × k [ x ] , ✇❤♦s❡ ❝♦❡✣❝✐❡♥ts A i ❛r❡ k × k ♠❛tr✐❝❡s✱ ✇❡ ❞❡✜♥❡ ✐ts ✈❛❧✉❡ ❛t ❛♥ ♦♣❡r❛t♦r R ∈ B ( H ) ❛s ❢♦❧❧♦✇s✿ p ( R ) := A 0 ⊗ I + A 1 ⊗ R + · · · + A t ⊗ R t ∈ C k ⊗ H, ✇❤❡r❡ I ✐s t❤❡ ✐❞❡♥t✐t② ♦♣❡r❛t♦r ❛♥❞     a 11 . . . a 1 n a 11 R . . . a 1 n R . . . . . .  ⊗ R := . .  .     . . . .   a m 1 . . . a mn a m 1 R . . . a mn R ❚❛t✐❛♥❛ ●✳ ●❡r❛s✐♠♦✈❛ ❯♥✐t❛r② s✐♠✐❧❛r✐t②✳ ◆❡✇ ❝r✐t❡r✐❛