Let $ (C,D)$ be a pair of bounded linear operators on a complex Hilbert space $ E$ . The Euclidean operator radius is defined by $ $ w_e(C,D)=\displaystyle\sup_{\|x\|=1}\left(|\langle Cx,x \rangle|^2+|\langle Dx,x \rangle|^2\right)^{1/2}.$ $ Moreover, the following inequality holds: $ $ \frac{\sqrt{2}}{4}\|C^*C+D^*D\|^{1/2}\leq w_e(C,D)\leq \|C^*C+D^*D\|^{1/2}.$ $

For the second inequality, the following example show that we can have equality:

Let $ (C,D)=(B,B)$ , with $ B=\begin{pmatrix}1&0\0&0\end{pmatrix}$ (operator on $ (\mathbb{C}^2,\|\cdot\|)$ ). Hence, I get $ w_e(C,D)=\sqrt{2}$ and $ \|C^*C+D^*D\|^{1/2}=\sqrt{2}.$

I facing difficulties to find a pair of operators $ (C,D)$ such that $ $ \frac{\sqrt{2}}{4}\|C^*C+D^*D\|^{1/2}= w_e(C,D).$ $

I try with the following example:

Let $ (C,D)=(\frac{1}{\sqrt{2}}A,\frac{1}{\sqrt{2}}A)$ , with $ A=\begin{pmatrix}0&0\1&0\end{pmatrix}$ (operator on $ (\mathbb{C}^2,\|\cdot\|)$ ). Hence, I get $ $ w_e(C,D)=w(A)=\frac{1}{2},$ $ but if I don’t make wrong in calculus I get $ $ \frac{\sqrt{2}}{4}\|C^*C+D^*D\|^{1/2}=\frac{\sqrt{2}}{4}.$ $

Thank you in advance.