• Journal of the Korean Mathematical Society
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• v.40 no.2
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• pp.307-317
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• 2003

For linear operators between Banach algebras "spectral boundedness" is derived from ordinary boundedness by substituting spectral radius for norm. The interplay between this concept and some of its near relatives is conspicuous in a result of Curto and Mathieu.

• Kyungpook Mathematical Journal
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• v.57 no.1
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• pp.125-132
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• 2017

We obtain commutativity-free characterizations of higher derivations on a unital Banach algebra that map into its Jacobson radical. We also investigate the spectral boundedness of generalized higher derivations.

• Kyungpook Mathematical Journal
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• v.53 no.4
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• pp.497-505
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• 2013

The noncommutative Singer-Wermer conjecture states that every derivation on a Banach algebra (possibly noncommutative) leaves primitive ideals of the algebra invariant. This conjecture is still an open question for more than thirty years. In this note, we approach this question via some sufficient conditions for the separating ideal of ${\phi}$-derivations to be nilpotent. Moreover, we show that the spectral boundedness of ${\phi}$-derivations implies that they leave each primitive ideal of Banach algebras invariant.

• Bulletin of the Korean Mathematical Society
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• v.47 no.1
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• pp.151-157
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• 2010

In this note, we obtain range inclusion results for left Jordan derivations on Banach algebras: (i) Let $\delta$ be a spectrally bounded left Jordan derivation on a Banach algebra A. Then $\delta$ maps A into its Jacobson radical. (ii) Let $\delta$ be a left Jordan derivation on a unital Banach algebra A with the condition sup{r$(c^{-1}\delta(c))$ : c $\in$ A invertible} < $\infty$. Then $\delta$ maps A into its Jacobson radical. Moreover, we give an exact answer to the conjecture raised by Ashraf and Ali in [2, p. 260]: every generalized left Jordan derivation on 2-torsion free semiprime rings is a generalized left derivation.