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 $T$.
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2$\begingroup$ 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. $\endgroup$– Jochen GlueckCommented May 30, 2022 at 16:33
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I think the example in Previous question in mathoverflow works both in the real and in the complex case.
answered May 30, 2022 at 8:51