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## Main Question or Discussion Point

Hi, can anyone give me an example of a bounded operator on a Hilbert space that has dense range but is not surjective? (Preferably on a separable Hilbert space)

Im pretty sure such an operator exists since the open mapping theorem requires surjectivity and not just dense range, but its just bothering me that I can't find an example.

Im pretty sure such an operator exists since the open mapping theorem requires surjectivity and not just dense range, but its just bothering me that I can't find an example.