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Abstract: In this paper functional equations related to derivations on semiprime rings and standard operator algebras are investigated. We prove, for example, the following result, which is related to a classical result of Chernoff. Let ▫$X$▫ be a real or complex Banach space, let ▫$L(X)$▫ be the algebra of all bounded linear operators of ▫$X$▫ into itself and let ▫$A(X) subset L(X)$▫ be a standard operator algebra. Suppose there exist linear mappings ▫$D,G colon A(X) to L(X)$▫ satisfying the relations ▫$D(A^3)=D(A^2)A + A^2G(A)$▫, ▫$G(A^3) = G(A^2)A + A^2D(A)$▫ for all ▫$A in A(X)$▫. In this case there exists ▫$B in L(X)$▫ such that ▫$D(A) = G(A) = [A,B]$▫ holds for all ▫$A in A(X)$▫.
Keywords: matematika, algebra, prakolobar, polprakolobar, Banachov prostor, standardna operatorska algebra, odvajanje, jordansko odvajanje, jordansko trojno odvajanje, mathematics, algebra, prime ring, semiprime ring, Banach space, standard operator algebra, derivation, Jordan derivation, Jordan triple derivation
Published in DKUM: 10.07.2015; Views: 1479; Downloads: 68
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Abstract: The purpose of this paper is to prove the following result. Let ▫$m ge 1$▫, ▫$n ge 1$▫ be some fixed integers and let ▫$R$▫ be a prime ring with ▫$text{char}(R)= 0$▫ or ▫$(m+n)^2 < text{char}(R)$▫. Suppose there exists an additive mapping ▫$T colon R to R$▫ satisfying the relation ▫$2(m+n)^2T(x^3) = m(2m+n)T(x)x^2 + 2mnxT(x)x + n(2n+m)x^2T(x)$▫ for all ▫$x in R$▫. In this case ▫$T$▫ is a two-sided centralizer.
Keywords: matematika, algebra, kolobar, prakolobar, polprakolobar, Banachov prostor, Hilbertov prostor, algebra vseh omejenih linearnih operatorjev, standardna operatorska algebra, odvajanje, jordansko odvajanje, centralizator, algebra, ring, prime ring, semiprime ring, Banach space, Hilbert space, algebra of all bounded linear operators, standard operator algebra, derivation, Jordan derivation, left (right) centralizer, two-sided centralizer, left (right) Jordan centralizer, (m, n)-Jordan centralizer
Published in DKUM: 10.07.2015; Views: 1422; Downloads: 140
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Abstract: Identities related to derivations on semiprime rings and standard operator algebras are investigated. We prove the following result which generalizes a classical result of Chernoff. Let ▫$X$▫ be a real or complex Banach space, let ▫$L(X)$▫ be the algebra of all bounded linear operators of ▫$X$▫ into itself and let ▫$A(X) subseteq L(X)$▫ be a standard operator algebra. Suppose there exists a linear mapping ▫$D:A(X) to L(X)$▫ satisfying the relation ▫$2D(A^{3}) = D(A^2)A + A^2D(A) + D(A)A^2 + AD(A^2)$▫ for all ▫$A in A(X)$▫. In this case ▫$D$▫ is of the form ▫$D(A) = AB-BA$▫ for all ▫$A in A(X)$▫ and some fixed ▫$B in L(X)$▫, which means that ▫$D$▫ is a linear derivation.
Keywords: matematika, algebra, prakolobar, polprakolobar, Banachov prostor, standardna operatorska algebra, odvajanje, jordansko odvajanje, jordansko trojno odvajanje, mathematics, algebra, prime ring, semiprime ring, Banach space, standard operator algebra, derivation, Jordan derivation, Jordan triple derivation
Published in DKUM: 10.07.2015; Views: 1598; Downloads: 131
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Abstract: The main purpose of this paper is to prove the following result. Let ▫$R$▫ be a prime ring of characteristic different from two and let ▫$T : R to R$▫ be an additive mapping satisfying the relation ▫$T(x^3) = T(x)x^{2} - xT(x)x + x^{2} T(x)$▫ for all ▫$x in R$▫. In this case ▫$T$▫ is of the form ▫$4T(x) = qx + xq$▫, where ▫$q$▫ is some fixed element from the symmetric Martindale ring of quotients. This result makes it possible to solve some functional equations in prime rings with involution which are related to bicircular projections.
Keywords: matematika, algebra, prakolobar, polprakolobar, funkcijska identiteta, odvajanje, jordansko odvajanje, involucija, bicirkularni projektor, mathematics, algebra, prime ring, semiprime ring, functional identity, derivation, Jordan derivation, involution, bicircular projection
Published in DKUM: 10.07.2015; Views: 1398; Downloads: 85
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Abstract: Naj bo ▫$K$▫ kolobar. Biaditivna simetrična preslikava ▫$D(.,.):K times K to K$▫ je simetrična biderivacija, če je za vsak fiksen ▫$y in K$▫ preslikava ▫$x mapsto D(x,y)$▫ derivacija. Glavni namen članka je dokazati rezultat v smislu klasičnega izreka E. Posnerja, ki pravi naslednje: Če je ▫$K$▫ prakolobar s karakteristiko različno od dva in sta ▫$D_1$▫ in ▫$D_2$▫ od nič različni derivaciji, potem preslikava ▫$x mapsto D_1(D_2(x))$▫ ne more biti derivacija.
Keywords: matematika, asociativni kolobarji in algebre, kolobar, prakolobar, polprakolobar, derivacija, simetrična biderivacija, mathematics, associative rings and algebras, prime ring, semiprime ring, derivation, simetric biderivation, semiprime ring, Banach algebra
Published in DKUM: 10.07.2015; Views: 1326; Downloads: 98
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