We describe all Rota—Baxter operators of any weight on the space of matrices from \(M_2(F)\) considered under the product \(a\circ b = (ab + ba)/2\) and usually denoted as \(M_2(F)^{(+)}\) . This algebra is known to be a simple Jordan one. We introduce symmetrized Rota—Baxter operators of weight \(\lambda \) and show that every Rota—Baxter operator of weight 0 on \(M_2(F)^{(+)}\) either is a Rota—Baxter operator of weight 0 on \(M_2(F)\) or is a symmetrized Rota—Baxter operator of weight 0 on the same \(M_2(F)\) . We also prove that every Rota—Baxter operator of nonzero weight \(\lambda \) on \(M_2(F)^{(+)}\) is either a Rota—Baxter operator of weight \(\lambda \) on \(M_2(F)\) or is, up to the action of \(\phi :R\rightarrow -R-\lambda \textrm{id}\) , a symmetrized Rota—Baxter operator of weight \(\lambda \) on \(M_2(F)\) .
