a question about contractions on Hilbert spaces Let $\cal{H}$ be a Hilbert space, $T_1,T_2\in\cal{B(H)}$, 


*

*$\|T_1(h_1)+T_2(h_2)\|^2\leq\|h_1\|^2+\|h_2\|^2$ for all $h_1,h_2\in\cal{H}$.

*$T_1T^\ast_1+T_2T^\ast_2\leq I$.
Then can we verify that 1 holds if and only if 2 holds ?

As Paul remarked above- this is not true. Take $h_1$ and $h_2$ to be two orthogonal vectors in $\mathcal{H}$, and $T_1=T_2=I$. Then, your condition 1) holds, but condition 2) doesn't.
