Stability for Jordan left derivations mapping into the radical of Banach algebras (Q2911456)

The article deals with the Ulam stability property for a ring Jordan left derivation \(d\) in Banach algebras. A ring Jordan left derivation is an additive mapping \(d:\;{\mathcal A} \to {\mathcal A}\) satisfying the condition \(d(x^2) = 2xd(x)\). Really it is considered a function \(f:\;{\mathcal A} \to {\mathcal A}\) satisfying NEWLINE\[NEWLINE\|f(x + y) - f(x) - f(y)\| \leq \varphi(x,y),NEWLINE\]NEWLINE NEWLINE\[NEWLINE\|f(xy + yx) - 2xf(y) - 2yf(x)\| \leq \phi(x,y),NEWLINE\]NEWLINE where \(\varphi,\phi:\;{\mathcal A} \times {\mathcal A} \to [0,\infty)\), NEWLINE\[NEWLINE\widetilde{\varphi}(x,y) = \sum_{k=0}^\infty \frac{\varphi(a^kx,a^ky)}{a^k} < \infty, \quad \lim_{n \to \infty} \;\frac{\phi(a^nx,y)}{a^n} = 0.NEWLINE\]NEWLINE It is proved that in this case there exists a unique ring Jordan left derivation \(d:\;{\mathcal A} \to {\mathcal A}\) such that NEWLINE\[NEWLINE\|f(x) - d(x)\| \leq \omega(x,y),NEWLINE\]NEWLINE where \(\omega(x,y)\) is determined by \(\varphi(x,y)\) and \(\phi(x,y)\). Also some variations and modifications of this statement are given.