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Abstract: The main purpose of this paper is to study the following: Let m, n, and $k_{i}, i = 1, 2, ..., n$ be positive integers and let $R$ be a $2m(m+ k_{1} + k_{2} + ... + k_{n} -1)!$-torsion free semiprime ring. Suppose that there exist derivations $D_{i} : R \to R, i = 1, 2, ..., n + 1$ , such that $D_{1}(x^{m})x^{k_{1}+...+k_{n}}+x^{k_{1}} D_{2}(x^{m})x^{k_{2}+...+k_{n}}+...+x^{k_{1}+...+k_{n}}D_{n+1}(x^{m})=0$ holds for all $x \in R$. Then we prove that $D_{1}+D_{2}+...+D_{n+1}=0$ and that the derivation $k_{1}D_{2}+(k_{1}+k_{2})D_{3}+...+(k_{1}+k_{2}+...+k{n})D_{n+1}$ maps $R$ into its center. We also obtain a range inclusion result of continuous derivations on Banach algebras.
Keywords: mathematics, algebra, associative rings and algebras, prime rings, Banach algebras, identities, derivations
Published in DKUM: 31.03.2017; Views: 1388; Downloads: 450
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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: 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: Naj bo ▫$H$▫ kompleksni Hilbertov prostor, ▫$mathcal{B}(H)$▫ algebra vseh omejenih linearnih operatorjev na ▫$H$▫ ter ▫$mathcal{A}(H)$▫ standardna operatorska algebra, zaprta za adjungiranje. Če je ▫$T: mathcal{A}(H) to mathcal{B}(H)$▫ linearna preslikava, ki zadošča identiteti ▫$T(AA^ast A) = T(A)A^ast A - AT(A^ast)A + AA^ast T(A)$▫ za vsak ▫$A$▫ iz ▫$mathcal{A}(H)$▫, potem je ▫$T(A) = AB + BA$▫ za vsak ▫$A$▫ iz A(H), kjer je ▫$B$▫ operator iz ▫$mathcal{B}(H)$▫.
Keywords: matematika, algebra, kolobar, prakolobar, polprakolobar, Banachov prostor, Hilbertov prostor, standardna operatorska algebra, odvajanje, jordansko odvajanje, bicirkularni projektor, mathematics, algebra, ring, ▫$^ast$▫-ring, prime ring, semiprime ring, Banach space, Hilbert space, standard operator algebra, derivation, Jordan derivation, bicircular projection
Published in DKUM: 10.07.2015; Views: 1406; Downloads: 98
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Keywords: matematika, algebra, neasociativne topološke algebre, integralski operatorji, Bernsteinovo jedro, Banach-Bernsteinova algebra, mathematics, algebra, nonassociative topological algebras, integral operators, Banach-Bernstein algebra, Bernstein kernel
Published in DKUM: 10.07.2015; Views: 967; Downloads: 33
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