On spectrum of a linear operator.

Question

Let $H$ be a Hilbert space and $S,T$ bounded linear operators defined on $H$. I am interested in showing that:

1. If $S,T$ are self-adjoint operators with compact $S$ and $ST=TS$ then $$\sigma(T+S)\subset \sigma(T)\cup \sigma_p(T+S).$$

Here $\sigma(T)$ denotes the spectrum of $T$ defined by: $$\sigma(T)=\{\lambda\in\mathbb{C} : T-\lambda I\hspace{.2cm}\mbox{it is not invertible}\}.$$ and $\sigma_p(T)$ is the set of eigenvalues of $T$.

Any contribution will be appreciated. Greetings!!

Answer 1

As written, this is not true. Let $H=L^2[0,1]\oplus L^2[2,3]$. Let $f$ be the identity function $f(t)=t$, and $$ T=M_f\oplus 0,\qquad S= 0\oplus M_f, $$ where $M_f$ is the multiplication operator. Then $T+S=M_f\oplus M_f$ has no eigenvalues, so the claimed inclusion becomes $\sigma(S+T)\subset\sigma(T)$. But $$ \sigma(S+T)=[0,1]\cup[2,3], $$ while $$ \sigma(T)=[0,1]. $$