Give an example of a non-self-adjoint operator on a Hilbert space $H$ whose range is $H$ and which is not invertible.

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Give an example of a non-self-adjoint operator on a Hilbert space $H$ whose range is $H$ and which is not invertible.

I cannot think of an example to save my life. Any solutions/hints are greatly appreciated.

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How about the shift map on $\ell^2(\mathbb{N})$?

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The canonical example would be the reverse shift on $\ell^2 (\mathbb N) $. That is, the operator $T $ given by $$T (a_1,a_2,\ldots)=(a_2,a_3,\ldots). $$