Local and 2-local Lie-type Derivations of Operator Algebras on Banach Spaces

Let X be a Banach space over the field \(\mathbb {F}\) ( \(\mathbb {F}\) is either the real field \(\mathbb {R}\) or the complex field \(\mathbb {C}\) ). Let B(X) be the set of all bounded linear operators on X and F(X) be the set of all finite rank operators in B(X). A subalgebra \(\mathcal {A}\) of B(X) is called a standard operator algebra if \(\mathcal {A}\) contains F(X). Suppose that \(\delta \) is a map from \(\mathcal {A}\) into B(X). Firstly, we prove that if \(\delta \) is a Lie-type derivation, then \(\delta \) has the standard form. Furthermore, we show that if \(\delta \) is a local Lie-type derivation, then \(\delta \) is a Lie-type derivation. Finally, we prove that if \(\delta \) is a 2-local Lie n-derivation, then \(\delta =d+\tau \) , where d is a derivation, and \(\tau \) is homogeneous map from \(\mathcal {A}\) into \(\mathbb {F}I\) such that \(\tau (A+B)=\tau (A)\) for each A, B in \(\mathcal {A}\) where B is a sum of \((n-1)\) -th commutators.