Thank you Delio. I initially thought it would just be a simple exercise that I somehow couldn't figure out so I posted it on MSE lest it be labeled as not "research level", but it seems deeper than I originally thought.

Thanks all for your comments. I have deleted my question on MSE to conform to this requirement.

Liviu: Thank you! Would you mind sharing with me one of the many examples where this fail in the Hilbert space? I am still trying to get some intuition about this.

Andres, thanks for your comment. What would you suggest if I am trying to maximize the exposure this question get? I had posted my questions at MSE for 48 hours with no response before I posted it here.

Hi Alex, Thanks for your reply. Two questions: a) Is the operator norm of "multiplied by $f$" operator bounded by its sup (or $L^\infty$ norm)? b) For the case of $f$ smooth and cptly supported, can we use the fact that $k(x, y)$ can be approximated in sup norm by $\sum_i g_i(x)h_i(y)$? If so, is the operator norm of the "difference kernel" bounded by its sup norm? Thanks!

Thanks for all your help above. It took me a while to figure out what $e_i \otimes e_j$ ("tensorial notation adopted by Schatten"?) means, blame my rusty functional analysis! Definition in p.2 of this article and the fact that all compact operators on Hilbert space are norm limit of finite rank operators are helpful for my understanding: plms.oxfordjournals.org/content/s3-17/1/115.full.pdf and jstor.org/stable/2374892?seq=4