Do you know an example of a linear bounded operator acting on a Banach (or even Hilbert) space whose residual spectrum is non-empty but the point and continuous spectrum are empty?


Take $X=l^{2}$. Now define the following continuous linear map

$$T:l^{2} \to l^{2}$$

$$T(x_{1},x_{2},.....)=(0, x_{1},\frac{x_{2}}{2}, \frac{x_{3}}{3} .... )$$

You can show that this $\sigma(T)=\{\ 0 \}\ $. And 0 is in the residual spectrum since T is one-one but $Range(T)$ is not dense in $l^{2}$.


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