Example of dense non surjective operator

[ak416]

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.

 

[dpsisi]

How about \ell_1 (all absolutely summable sequences) being mapped into L_2[0, 2\pi) ( all complex valued square integrable functions on [0, 2\pi) ) by

(a_n) -> \sum a_n \exp( \sqrt{-1} n x)