On Jordan $\sigma$-centralizers and related linear maps in algebras

Let $\mathcal{B}$ be an algebra over a commutative ring with identity $S$, and let $\sigma$: $\mathcal{B}\to \mathcal{B}$ be an algebra homomorphism. In this paper, we study linear operators $\mathrm{\Delta }$: $\mathcal{B}\to \mathcal{B}$ that are constrained by zero-product conditions involving the Jordan product $u\circ v=uv+vu$. In particular, we consider mappings that satisfy

$$uv=0\Rightarrow \mathrm{\Delta }(u\circ v)=\mathrm{\Delta }(u)\circ \sigma (v),uv=0\Rightarrow \mathrm{\Delta }(u\circ v)=\sigma (u)\circ \mathrm{\Delta }(v),$$

and

$$uv=0\Rightarrow \mathrm{\Delta }(u\circ v)=\mathrm{\Delta }(u)\circ \sigma (v)=\sigma (u)\circ \mathrm{\Delta }(v).$$

Assuming the endomorphism $\sigma$ is bijective, we prove that the scenario essentially simplifies to the identity case $\sigma ={\mathrm{i}\mathrm{d}}_{\mathcal{B}}$. Such a simplification enables a comprehensive the forms of these linear operators. As a result, we obtain precise expressions for these operators across diverse algebraic structures, including generalized matrix algebras, upper-triangular algebras, von Neumann algebras, standard operator algebras, and nest algebras. Moreover, this approach produces analogous results for Jordan $\sigma$-centralizers, thus extending and integrating various prior findings in the field.