Let $ \mathcal{B}(F)$ the algebra of all bounded linear operators on a complex Hilbert space $ F$ . Let $ S_1, S_2\in \mathcal{B}(F)$ . We define $ $ W(S_1,S_2)=\{(\langle S_1 y\; ,\;y\rangle,\langle S_2 y ,\;y\rangle):y \in F,\;\;\|y\|=1\}.$ $ I see in a paper that the cases in which $ W(S_1,S_2)$ is convex are the following:

$ (1)$ $ (S_1,S_2)$ is a pair of commuting normal operators.

$ (2)$ $ (S_1,S_2)$ is a pair of commuting operators in a two dimensional Hilbert spaces.

Assume that $ S_1S_2=S_2S_1$ . What do you think about the convexity of $ W(S_1,S_2)$ ? I try with many examples and I get always $ W(S_1,S_2)$ is convex. But I’m facing difficulties to prove that if $ S_1S_2=S_2S_1$ , then $ W(S_1,S_2)$ is always convex.

Thank you!!