Bulletin of the
Korean Mathematical Society BKMS

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$\{r(c^{-1}\delta(c)): c \in A \text{ invertible} \}< \infty$. 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.

Keywords: (generalized) left Jordan derivation, (generalized) left derivation, derivation, spectral boundedness, Jacobson radica

MSC numbers: 46H99, 47B47, 16N60