An inequality of operator norm

Question

I 'm proving $$ \max \left(\|A\|^{2},\|B\|^{2}\right) \leq\|T\|^{2} \leq\|A\|^{2}+\|B\|^{2} $$ where $ T=A+i B $ is cartesian decomposition of T. Since $\|Tx\|^{2}= \|Ax\|^{2} + \|Bx\|^{2}$,I'm wondering whether this $$ \max \left(\|A\|^{2},\|B\|^{2}\right) \leq \frac{\|Ax\|^{2} + \|Bx\|^{2}}{\|x\|^2} $$ is true. Any help is appreciated.

Answer 1

Surely false if your inequality is supposed to be for each fixed $x$. We can have $Ax=0=Bx$ without $A$ and $B$ being zero.

However the inequality is true if you have sup over all non-zero $x$ on RHS.