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 $ \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 \Delta(u\circ v) = \Delta(u)\circ \sigma(v), \; \; \; uv = 0 \Rightarrow \Delta(u\circ v) = \sigma(u)\circ \Delta(v), $ and$ uv = 0 \Rightarrow \Delta(u\circ v) = \Delta(u)\circ\sigma(v) = \sigma(u)\circ\Delta(v). $Assuming the endomorphism $ \sigma $ is bijective, we prove that the scenario essentially simplifies to the identity case $ \sigma = \mathrm{id}_{\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.