Let $E$ be a complex Hilbert space, with inner product $\langle\cdot\;| \;\cdot\rangle$ and the norm $\|\cdot\|$ and let $\mathcal{B}(E)$ the algebra of all bounded linear operators from $E$ to $E$.

I want to show that for $(T_1,...,T_d) \in \mathcal{B}(E)^d$ we have: $$\displaystyle\frac{1}{2}\left\|\displaystyle\sum_{k=1}^dT_k^*T_k \right\|^{1/2}\leq \omega(T_1,\cdots,T_d) \leq \left\|\displaystyle\sum_{k=1}^dT_k^*T_k \right\|^{1/2},$$ where $$\omega(T_1,\cdots,T_d)=\displaystyle\sup_{\|x\|=1}\bigg(\displaystyle\sum_{k=1}^d|\langle T_kx\;|\;x\rangle|^2\bigg)^{1/2}.$$

And you for you help.