Inner product and adjoint mapping in matrices

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Let $\mathcal{A}_1:S^n \to S^n$ be defined by $\mathcal{A}_1(P)=-(A^*P+PA)$. How can I find $\mathcal{A}_1^{adj} (Z)$, where $\mathcal{A}_1^{adj}:S^n \to S^n$?