Commutators and images of noncommutative polynomials
Brešar, Matej
Naj bo ▫$A$▫ algebra in ▫$f$▫ nekonstanten nekomutativen polinom. V prvem delu članka obravnavamo zvezo med ▫$[A,A]$▫, linearno lupino vseh komutatorjev v ▫$A$▫, in span▫$\,f(A)$▫, linearno lupino ...slike ▫$f$▫ v ▫$A$▫. Med drugim pokažemo, da iz ▫$[A,A]=A$▫ sledi span▫$\,f(A)=A$▫. V drugem delu izpeljemo rezultate Waringovega tipa za slike polinomov. Tako dokažemo, da se v primeru, ko je ▫$A$▫ matrična algebra ▫$M_n(C)$▫, kjer je ▫$C$▫ komutativna enotska algebra nad poljem ▫$F$▫ s karakteristiko ▫$0$▫, in polinom ▫$f$▫ ni identiteta ali centralni polinom ▫$M_n(F)$▫, vsak komutator v ▫$A$▫ da zapisati kot razlika dveh elementov, ki sta vsoti ▫$7788$▫ elementov iz ▫$f(A)$▫ (če je ▫$C=F$▫ algebraično zaprto polje, zadoščajo 4 elementi). Podobni rezultati so dobljeni za nekatere druge algebre, med drugim za algebro zveznih linearnih operatorjev na Hilbertovem prostoru.
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