  A well-known fact is that there does not exist any compatible left-symmetric
structures on a finite-dimensional complex semisimple Lie algebra (see
\cite{Chu1974}). This result is not valid in semisimple Lie superalgebra case.
In this paper, we study the compatible Left-symmetric superalgebra (LSSA for
short) structures on complex simple Lie superalgebras. We prove that there is
not any compatible LSSA structure on a finite-dimensional complex simple Lie
superalgebra except for the classical simple Lie superalgebra $A(m,n)(m\neq n)$
and Cartan simple Lie superalgebra $W(n)(n\geq 3)$. We also classify all
compatible LSSAs with a right-identity on A(0,1).
