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Let $T\in\mathscr{B(\mathcal{H})}$ and $X\in M_n(\mathbb{C})$. Is the following statement true? If $W(B\otimes X)\subseteq W(B\otimes T)$ for any $B\in M_n$ then $W(B\otimes (X+I_n))\subseteq W(B\otimes (T+I))$ for any $B\in M_n$ where $W(T):=\{\langle Tx,x\rangle: \Vert x\Vert=1\}$ is called the numerical range of $T$.

Comments: I have tried this in various ways. I have observed yet that if $W(T)$ be a straight line (both degenerate and nondegenerate case) or $W(T)$ contains a neighbourhood around $0$, the previous statement is true. But for other cases, I could neither yet prove the statement nor find a counterexample to disprove this. Any comment is highly appreciated. Thanks in advance.

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  • $\begingroup$ $X=0$ and $T=-I$ is a counter example, but I wouldn't be surprised if you want to rule this out? $\endgroup$
    – Jamie Gabe
    Commented Nov 23, 2019 at 22:58
  • $\begingroup$ @JamieGabe, How does it provide a counter example? As $W(B\otimes X)\subseteq W(B\otimes T)$ should be true for any $B\in M_n$ but according to your example, this is true for those $B$ with $0\in W(B)$. In other words, I wanted to check whether the following is true: $$\cap_{B\in M_n}\{X\in M_n: W(B\otimes X)\subseteq W(B\otimes T)\}\subseteq \cap_{B\in M_n}\{X\in M_n: W(B\otimes (X+I))\subseteq W(B\otimes (T+I))\}.$$ ofcourse this is not true for a perticular choice of $B$. $\endgroup$
    – Piku
    Commented Nov 24, 2019 at 6:57
  • $\begingroup$ @JamieGabe, Moreover if $T$ is normal then the statement is true as $W(A\otimes B)=\text{ Conv }W(A)W(B)$ whenever at least one of $A$ and $B$ is normal. $\endgroup$
    – Piku
    Commented Nov 24, 2019 at 9:26
  • $\begingroup$ Ah, sorry about that, it's definitely not a counterexample. My bad! $\endgroup$
    – Jamie Gabe
    Commented Nov 24, 2019 at 9:46
  • $\begingroup$ @JamieGabe, I think your example serves as a counterexample because of the following. Note that $W(B\otimes X)\subseteq W(B\otimes T)$ for any $B$ is equivalent with $W((\alpha B+\beta)\otimes X)\subseteq W((\alpha B+\beta)\otimes T)$ for any $\alpha,\beta\in\mathbb{C}, B\in M_n$ which allows us to take $0\in W(B)$ without loss of generality and then your example provides a counterexample. Is this argument correct? I apologize for making the statement that it is true for normal $T$ as I have realized now that the proof what I was thinking is not correct. $\endgroup$
    – Piku
    Commented Nov 24, 2019 at 10:07

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