If $W_0(U^*T_1U,U^*T_2U)= \emptyset$. Why $W_0(T_1,T_2)= \emptyset$?Where $U$ is unitary operator.

Question

Let $\mathcal{H}$ be a complex Hilbert space. Let $T_1,T_2\in \mathcal{B}(\mathcal{H})$. Let \begin{eqnarray*} W_0(T_1,T_2) &=&\{(\lambda_1,\lambda_2)\in \mathbb{C}^2;\;\exists\,(x_n)_n;\;\|x_n\|=1,\;(\langle T_1 x_n\; |\;x_n\rangle,\,\langle T_2 x_n\; |\;x_n\rangle)\to (\lambda_1,\lambda_2),\\ &&\phantom{++++++++++}\;\hbox{and}\;\displaystyle\lim_{n\rightarrow+\infty}(\|T_1x_n\|^2+\|T_2x_n\|^2)\rightarrow \|T_1\|^2+\|T_2\|^2\;\}. \end{eqnarray*}

Assume that $W_0(U^*T_1U,U^*T_2U)= \emptyset$. Why $W_0(T_1,T_2)= \emptyset$? Where $U$ is an unitary operator (i.e. $U^*U=UU^*=I$).

Thank you!

Answer 1

If $W_0(T_1,T_2) \neq \emptyset$ choose $\{x_n\}$ as in the definition. Let $y_n=U^{*}$$x_n$. Use this sequence to get a contradiction.