Similarity of two self-adjoint operators 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.

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). 
