Weak $(p,q)$-Jordan centralizer and derivation on rings and algebras

In the present paper, the authors discuss two new concepts that will be known as a weak $(p,q)$-Jordan centralizer and a weak $(p,q)$-Jordan derivation on an arbitrary ring $R$ and they prove that every weak $(p,q)$-Jordan derivation is a derivation on any prime ring $R$. Furthermore, every weak $(p,q)$-Jordan centralizer is a centralizer on a semiprime ring $R$. Later, they discuss the continuity of weak $(p,q)$-Jordan centralizer on a semisimple Banach algebra and prove that every weak $(p,q)$ -Jordan centralizer on a semisimple Banach algebra is a linear continuous operator. Moreover, these results are validated with actual examples.