# Proof verification related to a rotation of an operator and convexity

Let $$(\mathcal{H}, \langle\cdot,\cdot\rangle)$$ be a complex Hilbert space and $$\mathcal{B}(\mathcal{H})$$ is the algebra of all bounded linear operators on $$\mathcal{H}$$.

For $${\bf T} = (T_1,\cdots,T_n)\in \mathcal{B}(\mathcal{H })^n$$, we consider the following subset $$\mathbb{C}^n$$ $$\begin{eqnarray*} JtMaxW({\bf T}) &=&\{(\lambda_1,\cdots,\lambda_n)\in \mathbb{C}^n:\;\exists\,(x_i)\subset \mathcal{H}\;\;\hbox{such that}\;\|x_i\|=1,\displaystyle\lim_{i\rightarrow+\infty}\langle T_k x_i,x_i\rangle=\lambda_k,\\ &&\phantom{++++++++++}\;\hbox{and}\;\displaystyle\lim_{i\rightarrow+\infty}\|T_kx_i\|\rightarrow \|T_k\|,\; 1\leq k \leq n\;\}. \end{eqnarray*}$$

Why the following facts which is taken from this reference(theorem 3.8) hold? Note that Lemma 3.4 indicates that $$\pi_k(JtMaxW_A({\bf T}))$$ is convex for all $$k\in\{1,\cdots,n\}$$, where $$\pi_k$$ is the projection from $$\mathbb{C}^n$$ to $$\mathbb{C}$$ i.e. $$\pi_k(x_1,\cdots,x_n)=x_k$$, for all $$k\in\{1,\cdots,n\}$$.

If the above claim is false. Is it true if $$n=1$$?

My attempt: Suppose now that $$0\notin JtMaxW({\bf T})$$, then $$\forall\{x_i\}\in\mathcal{H}$$ such that $$\|x_i\|=1$$ we have, $$\langle T_k x_i,x_i\rangle\nrightarrow 0 \quad \mbox{or}\quad\|T_kx_i\|\nrightarrow \|T_k\|,\,\, 1\leq k \leq n.$$ Let us rotate $${\bf T}$$, that is we consider $${\bf \widehat{T}}=(e^{i\alpha_1}T_1,...,e^{i\alpha_n}T_n)$$. Moreover, if $$\lambda=(\lambda_1,...,\lambda_n)\in JtMaxW({\bf \widehat{T}})$$, then there exists a sequence $$\{y_i\}$$ in $$\mathcal{H}$$ such that $$\|y_i\|=1\quad \mbox{and}\quad\|\widehat{T}_ky_i\|\rightarrow \|\widehat{T}_k\|,\,\, 1\leq k \leq n.$$ i.e. $$\|y_i\|=1,\,\,e^{-i\alpha_k}\langle T_k y_i,y_i\rangle\rightarrow \lambda_k \quad \mbox{and}\quad\|T_ky_i\|\rightarrow \|T_k\|,\,\, 1\leq k \leq n.$$ Since, $$\displaystyle\lim_{i\rightarrow\infty}\langle T_k y_i,y_i\rangle\neq0$$, then $$\lambda_k\neq0$$ for all $$1\leq k \leq n.$$ Assume that there exists $$\delta_k$$ such that $$\Re e(\lambda_k)\geq\delta_k>0$$

Why for all $$x\in \mathcal{H},\;\|x\|=1$$ we have $$\Re e\langle \widehat{T}_k x,x\rangle\geq\delta_k>0$$)?

The claim as it stands is false because $$JtMaxW(T)$$ might be empty. Then clearly, $$(0,\ldots,0)\notin JtMaxW(T)$$, but also $$JtMaxW(T') = \emptyset$$ for any "rotation" $$T'$$ of $$T$$.
An example for which $$JtMaxW(T)$$ is empty is given by $$T_1 = \begin{pmatrix}1&0\\0&0\end{pmatrix}\quad\text{and}\quad T_2 = \begin{pmatrix}0&0\\0&1\end{pmatrix}.$$