Convexity of a special kind of set

Question

Let $\mathcal{H}$ be a separable Hilbert space and $T\in B(\mathcal{H})$. Is the following set convex \begin{align} \{\langle Te_1,e_2\rangle:\{e_1,e_2\} \text{ are orthonormal set in } \mathcal{H}\}? \end{align}

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Could you please suggest to me some references in case this exists in the literature? I am not able to show it on my own.

Answer 1

This is not true. Take $$ T = \pmatrix{ 0 & 1\\ -1& 0}. $$ Let $e_1=\pmatrix{x_1\\x_2}$, $e_2=\pm \pmatrix{-x_2\\x_1}$, then $$ \langle Te_1,e_2\rangle = \pm 1, $$ so the set in question is equal to $\{-1,+1\}$, which is not convex.