• 대한수학회보
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• 제38권4호
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• pp.709-718
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• 2001

In this paper we show that if D and G are continuous linear Jordan derivations on a Banach algebra A satisfying [D(x), x]x - x[G(x),x] $\epsilon$ rad(A)for all $\epsilon$ A, then both D and G map A into rad(A).

• 대한수학회보
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• 제37권3호
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• pp.429-435
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• 2000

Our main goal is to show that if there exist Jordan derivations D, E and G on a noncommutative 2-torsion free prime ring R such that$(G^2(x)+E(x))D(x)=0\ or\ D(x)(G^2(x)+E(x))=0\ for\ all\ x\inR$, then we have D=o or E=0, G=0.

• Journal of applied mathematics & informatics
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• 제6권1호
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• pp.303-307
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• 1999

Let A and B be two strictly dense Banach Algebras on X and Y respectively where X and Y are Banach space. We give some conditions under which full spectrum preserving linear mappings from A into B Jordan morphisms and X is homomorphic to Y.

• 충청수학회지
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• 제29권4호
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• pp.523-530
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• 2016

Let R be a 3!-torsion free semiprime ring, and let $D:R{\rightarrow}R$ be a Jordan derivation on a semiprime ring R. In this case, we show that [D(x), x]D(x) = 0 if and only if D(x)[D(x), x] = 0 for every $x{\in}R$. In particular, let A be a Banach algebra with rad(A). If D is a continuous linear Jordan derivation on A, then we see that $[D(x),x]D(x){\in}rad(A)$ if and only if $[D(x),x]D(x){\in}rad(A)$ for all $x{\in}A$.

• 대한수학회논문집
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• 제28권3호
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• pp.535-558
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• 2013

The purpose of this paper is to prove that the noncommutative version of the Singer-Wermer Conjecture is affirmative under certain conditions. Let A be a noncommutative Banach algebra. Suppose there exists a continuous linear Jordan derivation $D:A{\rightarrow}A$ such that $D(x)^3[D(x),x]{\in}rad(A)$ for all $x{\in}A$. In this case, we show that $D(A){\subseteq}rad(A)$.

• Journal of applied mathematics & informatics
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• 제13권1_2호
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• pp.425-432
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• 2003

Our main goal is to show that if there exists a continuous linear Jordan derivation D on a noncommutative Banach algebra A such that n$^{x}$ D(x)n+xD(x)x$^{n}$ $\in$ rad(A) for all x $\in$ A, then D maps A into rad(A).

• 대한수학회보
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• 제32권2호
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• pp.367-372
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• 1995

In 1955 Singer and Wermer [12] proved that every bounded derivation on a commutative Banach algebra maps into its radical. They conjectured that the continuity of the derivation in their theorm can be removed. In 1988 Thomas [13] proved their conjecture ; Every derivation on a commutative Banach algebra maps into its radical. For noncommutative versions, in 1984 B. Yood [15] proved that the continuous derivations on Banach algebras satisfing [D(a),b] $\in$ Rad(A) for all a, b $\in$ A have the radical range, where [a,b] will be denote the commutator ab-ba. In 1990 M.Bresar and J.Vukman [1] have generlized Yood's result, that is, the continuous linear Jordan derivation on Banach algebra that satisfies [D(a),a] $\in$ Rad(A) for all a $\in$ A has the radical range. In next year Mathieu and Murphy [5] proved that every bounded centralizing derivation on Banach algebras has its image in the radical. Mathieu and Runde [6] removed the boundedness of that.

• 충청수학회지
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• 제14권1호
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• pp.21-28
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• 2001

In this paper we show that the following results remain valid for arbitrary Jordan derivations as well: Let d be a derivation of a complex Banach algebra A. If $d^2(x){\in}rad(A)$ for all $x{\in}A$, then we have $d(A){\subseteq}rad(A)$ ([5, p. 243]), and in a case when A is unital, $d(A){\subseteq}rad(A)$ if and only if sup{$r(z^{-1}d(z)){\mid}z{\in}A$ invertible} < ${\infty}$([3]), where rad(A) stands for the Jacobson radical of A, and r(${\cdot}$) for the spectral radius.

• 대한수학회논문집
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• 제17권4호
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• pp.607-618
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• 2002

Let A be a noncommutative Banach algebra. Suppose there exists a continuous linear Jordan derivation D : A \longrightarrowA such that D($\chi$)[D($\chi$),$\chi$]D($\chi$) $\in$ rad(A) for all $\chi$ A. In this case, we have D(A) ⊆ rad(A).

• 대한수학회보
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• 제39권2호
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• pp.211-224
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• 2002

In this paper we show that if D is a continuous linear Jordan derivation on a Banach algebra A satisfying [[D($x^{n}$), $x^{n}$, $x^{n}$] $\in$ rad(A) for a positive integer n and for all x${\in}$A, then D maps A into rad(A).