For a Borel measure $\mu$ define $\langle S_\mu x,y\rangle=\int_H\langle x,z\rangle \langle y,z\rangle \mu(z)$. An exercise in my book that I am reading says that I could find a $\mu$ s.t. $S_\mu$ is a bounded operator but not compact. If $\dim(H)<\infty$ then this is impossible. If $\dim(H)=\infty$, then the identity operator $I$ is not compact so assuming I can find a $\mu$ s.t. $S_\mu=I$, then that should suffice. To be clear this is an exercise in my book but is not a homework question for a class. Also, assume $H$ is real and separable Hilbert space.