A bounded operator acting on a complex Banach space has non-empty spectrum, and the proof of this fact uses the completeness of the space.

Is there any example of bounded operator acting on a complex non-complete normed space with empty spectrum?

A bounded operator acting on a complex Banach space has non-empty spectrum, and the proof of this fact uses the completeness of the space.

Is there any example of bounded operator acting on a complex non-complete normed space with empty spectrum?

I understand that the spectrum of an operator $T$ is the set of all complex numbers $\lambda$ such that $T-\lambda I$ is not bijective.