  Based on Johnson's operator formula for the equivariant Gromov-Witten theory
of $\mathbb{P}^1$-orbifolds, we give a new approach to the operator formalism
by Okounkov and Pandharipande regarding the $\mathbb{C}^*$-equivariant
Gromov-Witten theory of $\mathbb{P}^1$ relative to one or two points. We also
extend their operator formalism in the non-equivariant specialization to the
case where we allow negative contact orders.
