# Empty spectrum of an operator

Recently, I was studying the spectrum of an operator $$T$$ on a normed linear space $$X$$. In order to ensure that the spectrum is non-empty, one needs to assume that $$X$$ is a complex Banach space. But the books don't give sufficient justification. I would like to have an example of a real incomplete normed linear space as well as an example of a complex incomplete normed linear space $$X$$ and a bounded operator $$T:X \rightarrow X$$ such that the spectrum of $$T$$ is empty. In order to avoid confusion, I would like to define the spectrum: A scalar $$\lambda$$ is called a spectral value of a bounded operator $$T$$ on a normed linear space $$X$$ if $$T - \lambda I$$ is not invertible. '$$I$$' is the identity operator on $$X$$ and the set of all spectral values of $$T$$ is called the spectrum of $$T$$.

• I don't understand the point about the non-complete example over the real field. Even on the complete real space $\mathbb{R}^2$ there are simple operators with empty spectrum. May 30, 2022 at 16:33