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In this question $E$ stands for a Hilbert space over $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$.

For $(T_1,...,T_d) \in \mathcal{L}(E)^d$, is the following inequalities true ? $$\displaystyle\frac{1}{2}\left\|\displaystyle\sum_{k=1}^dT_k^*T_k \right\|^{1/2}\leq \omega(T_1,\cdots,T_d) ?,$$ where $$\omega(T_1,...,T_d)=\displaystyle\sup_{\|x\|=1}\bigg(\displaystyle\sum_{k=1}^d|\langle T_kx\;|\;x\rangle|^2\bigg)^{1/2}.$$

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If the operators $T_k$ are all non-zero and skew-symmetric over $\mathbb R$, then $\omega()$ is zero, and the desired inequality does not hold.

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