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.

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    $\begingroup$ You should use \langle and \rangle instead of < and >, respectively. $\endgroup$ – Cameron Williams Feb 11 '14 at 13:53
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    $\begingroup$ Take $H=\ell^2(\mathbb N)$ and $\mu=\sum_1^\infty \delta_{e_k}$, where $(e_k)_{k\in\mathbb N}$ is the "canonical basis" of $\ell^2$. $\endgroup$ – Etienne Feb 11 '14 at 15:12

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