Spectrally bounded φ-derivations on Banach algebras

Applying the density theorem on algebras with phi-derivations, we show that if a phi-derivation delta of a unital Banach algebra A is spectrally bounded, then [delta(A), A] subset of or equal to rad(A). Also, delta(A) subset of or equal to rad(A) if and only if sup{r(z(-1)delta(z)) | z is an element of A is invertible} < ∞, where r(a) denotes the spectral radius of a ∈ A.