Jordan left {g, h}-derivations over some algebras

Abstract: In this article, left {g, h}-derivation and Jordan left {g, h}-derivation on algebras are introduced. It is shown that there is no Jordan left {g, h}-derivation over $\mathcal{M}n(C)$ and $\mathbb{H}{\mathbb{R}}$, for g not equal to h. Examples are given which show that every Jordan left ${g, h}$-derivation over $\mathcal{T}n(C)$, $\mathcal{M}_n(C)$ and $\mathbb{H}{\mathbb{R}}$ are not left ${g, h}$-derivations. Moreover, we characterize left ${g, h}$-derivation and Jordan left ${g, h}$-derivation over $\mathcal{T}n(C)$, $\mathcal{M}_n(C)$ and $\mathbb{H}{\mathbb{R}}$ respectively. Also, we prove the result of Jordan left ${g, h}$-derivation to be a left ${g, h}$-derivation over tensor products of algebras as well as for algebra of polynomials.