Point spectrum

Remarks

Here $\ker$ is the kernel. The point spectrum is a subset of the spectrum $\sigma(T)$, and for $X = \C^n$ the two are equal.

Example: let $L(x_1, x_2, \ldots) = (x_2, x_3, \ldots)$ be the left shift on $\ell^2$. For $\abs{\lambda} < 1$, the vector $x = (1, \lambda, \lambda^2, \ldots)$ is in $\ell^2$ and $Lx = \lambda x$, so $\lambda \in \sigma_p(L)$. Conversely $Lx = \lambda x$ forces $x_{n+1} = \lambda x_n$, so $x = x_1 (1, \lambda, \lambda^2, \ldots)$, which is in $\ell^2$ with $x \ne 0$ only if $\abs{\lambda} < 1$; thus $\sigma_p(L)$ is the open unit disc.

Non-example: for the diagonal operator $D(x_n) = (x_n / n)$ on $\ell^2$, $0 \in \sigma(D)$ but $0 \notin \sigma_p(D)$, since $Dx = 0$ forces $x = 0$.

The point spectrum need not be closed, and it can be empty, as for the right shift on $\ell^2$.