Let H be a separable infinite-dimensional Hilbert space. Given the operators A ∈ B(H) and B ∈ B(H), we define M X := A X 0 B where X ∈ S(H) is a self-adjoint operator. In this paper, a necessary and sufficient condition is given for M X to be a left (right) Weyl operator for some X ∈ S(H). Moreover, it is shown that X∈S(H) σ ⋆ (M X) = X∈S(H)∩Inv(H) σ ⋆ (M X) = X∈B(H) σ ⋆ (M X) ∪ ∆, where σ * is the left (right) Weyl spectrum. Finally, we further characterize the perturbation of the left (right) Weyl spectrum for Hamiltonian operators.