Remark on square roots of self-adjoint weighted composition operators on H2

In this paper, we study square roots of self-adjoint weighted composition operators on $H^2$. More precisely, we focus on square roots $W_{f,\phi }$ of a self-adjoint operator $W_{g,\psi }=W_{f,\phi }^2$ when φ is a linear fractional selfmap of $D$. We also investigate several properties of such $W_{f,\phi }$. Finally, we show that the square roots $W_{f,\phi }$ may be other, nonself-adjoint weighted composition operators and have nontrivial invariant subspaces.