On the Generalized Jordan-Derivation Over -Paranormal Operators

Let B(H) be the set of all bounded linear operations on the separable infinite dimensional Hilbert spaces. The mapping $J_{A, B}(T)=TA-BT^*$ is called generalized Jordan*-derivations. Researchers have thoroughly investigated the characteristics of $J_{A, B}(T)$ within specific classes of operators A and B. The characterization of the properties of A and B relies significantly on the range of $J_{A, B}(T)$. In this paper, we investigate the generalized Jordan *-derivations $J_{A, B}(T)=TA-BT^*$ over *- paranormal operators A and B. The author shows among other results that if the identity operator belongs to the range of generalized Jordan *-derivations $J_{A, B}$ over *- paranormal operators A and B where $\partial\sigma(A)=\partial\sigma(B)$, then A and B are invertible skew-Hermitian operators.