Transposed Poisson structures on the extended Schrödinger-Virasoro and the original deformative Schrödinger-Virasoro algebras

We compute $\frac{1}{2}$-derivations on the extended Schrödinger-Virasoro algebras and the original deformative Schrödinger-Virasoro algebras. The extended Schrödinger-Virasoro algebras have neither nontrivial $\frac{1}{2}$-derivations nor nontrivial transposed Poisson algebra structures. We demonstrate that the original deformative Schrödinger-Virasoro algebras have nontrivial $\frac{1}{2}$-derivations, indicating that they possess nontrivial transposed Poisson structures.