What is the square-root of the nonnegative self-adjoint operator $L^2\to L^2,f\mapsto\int f$?

Question

Let $(E,\mathcal E,\mu)$ be a probability space. Since $\mathbb R$ is naturally embedded into $L^2(\mu)$, we may consider the operator $$U:L^2(\mu)\to L^2(\mu)\;,\;\;\;f\mapsto\int f\:{\rm d}\mu.$$ Obviously, $U$ is nonnegative and self-adjoint. Thus, there is a unique square-root $Q^{1/2}:L^2(\mu)\to L^2(\mu)$. But what's the explicit form of $Q^{1/2}$?

Answer 1

The integral of a constant with respect to a probability measure is itself. So $U^2 = U$ and thus $U$ is its own square root.