Classification of spectrum in functional analysis.

Question

As we know, the spectrum of an operator $T$ has a standard decomposition into three parts:

1. a point spectrum, consisting of eigenvalues of $T$ ;
2. a continuous spectrum, consisting of the scalars that are not eigenvalues but make the range of $T-\lambda$ a proper dense subset of the space;
3. a residual spectrum, consisting of all other scalars in the spectrum.

My question is are there any spectrum has continuous spectrum but no residual spectrum? Or conversely, a spectrum has residual spectrum but no continuous spectrum?

Answer 1

We can look at this perhaps from a different point of view that will make things more clear to you.

We can define the spectrum $\sigma(T)$ of a bounded linear operator $T$ on a Hilbert space $H$ to be the set of all $\lambda \in \mathbb{C}$ such that the $(T- \lambda I)$ is not a bijection.

• The point spectrum of $T$ consists of all $\lambda \in \sigma(T)$ such that $(T- \lambda I)$ is not one-to-one. In this case $\lambda$ is called an eigenvalues of $T$.

• The continuous spectrum of $T$ consists of all $\lambda \in \sigma(T)$ such that $(T- \lambda I)$ is one-to-one but not onto, and $range(T − \lambda I )$ is dense in $H$.

• The residual spectrum of A consists of all $\lambda \in \sigma(T)$ such that $(T- \lambda I)$ is one-to-one but not onto, and $range(T- \lambda I)$ is not dense in $H$.

Now I think you can construct the examples or understand the examples given by @user10354138 on your own.