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Can the same operator when defined on two different spaces have different spectra? For example and operator defined on $C_0$ and on $\ell_2$?

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Yes, there are quite a lot easy examples, e.g. consider the momentum operator $-i\frac{d}{dx}$ on $L^2(-a,a)$ where $a$ is some finite number and the Sobolev Space $H^1$ as domain. In this case, $e^{ikx}$ is an eigenfunction for every $k$ and therefore the spectrum is whole $\mathbb{C}$. However, if you choose $H^1$-functions with periodic boundary conditions as domain, this operator will be self-adjoint and only admit real eigenvalues.

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Yes, I have seen cases where an operator on $L^1(R^3,d^3x)$ has a much larger spectrum than on $L^2(R^3,d^3x)$. If my memory serves me well this was the case for a Fokker-Planck operator, the $L^2$ spectrum was real but the $L^1$ spectrum contained contributions outside the real axis.

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