Let $(X, \mu)$ be a finite measure space and let $A$ be a non-negative self-adjoint operator which generates a contraction semigroup $e^{tA}$ on $L^2(X, \mu)$. If additionally, we have that $e^{tA}$ is contractive on $L^\infty(X, \mu)$, is it true that $e^{tA}$ is contractive on $L^p(X, \mu)$, where $1 \leq p \leq \infty$?

Let $(X, \mu)$ be a finite measure space and let $A$ be a non-negative self-adjoint operator which generates a contraction semigroup $e^{tA}$ on $L^2(X, \mu)$. If additionally, we have that $e^{tA}$ is bounded on $L^\infty(X, \mu)$, is it true that $e^{tA}$ is bounded on $L^p(X, \mu)$, where $1 \leq p \leq \infty$?