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Conformal classical Yang-Baxter equation and $S$-equation naturally appear inthe study of Lie conformal bialgebras and left-symmetric conformal bialgebras.In this paper, they are interpreted in terms of a kind of operators, namely,$\mathcal O$-operators in the conformal sense. Explicitly, the skew-symmetricpart of a conformal linear map $T$ where $T_0=T_\lambda\mid_{\lambda=0}$ is an$\mathcal O$-operator in the conformal sense is a skew-symmetric solution ofconformal classical Yang-Baxter equation, whereas the symmetric part is asymmetric solution of conformal $S$-equation. One byproduct is that a finiteleft-symmetric conformal algebra which is a free $\mathbb{C}[\partial]$-modulegives a natural $\mathcal O$-operator and hence there is a construction ofsolutions of conformal classical Yang-Baxter equation and conformal$S$-equation from the former. Another byproduct is that the non-degeneratesolutions of these two equations correspond to 2-cocycles of Lie conformalalgebras and left-symmetric conformal algebras respectively. We also give afurther study on a special class of $\mathcal{O}$-operators called Rota-Baxteroperators on Lie conformal algebras and some explicit examples are presented.

Conformal classical Yang-Baxter equation, $S$-equation and $\mathcal{O}$-operators