Similarity of two self-adjoint operators

Question

I am wondering whether the following is correct:

Let $A$ and $B$ be two bounded self-adjoint, positive and invertible linear operators such that $\sigma(A)=\sigma(B)$ and $AB=BA$. Can we say that $A$ is necessarily similar to $B$?

I just couldn't find an answer.

Thanks.

Math.

Answer 1

Even if $\sigma(A)=\sigma(B)$, similarity sees multiplicity. For instance let $$ A=\begin{bmatrix} 1&0&0\\ 0&1&0\\ 0&0&2\end{bmatrix},\ \ \ B=\begin{bmatrix} 1&0&0\\ 0&2&0\\0&0&2\end{bmatrix}. $$ Then $AB=BA$, $\sigma(A)=\sigma(B)=\{1,2\}$, but they are not similar (for instance, because they don't have the same trace).