Showing that an operator is positive

Question

Let $\mathcal{H}$ be a Hilbert space, and $T$ an operator. The task is to prove that the corresponding self adjoint operator $ \begin{bmatrix} 1 & T \\ T^{*} & 1\\ \end{bmatrix} $ defined on $\mathcal{H}\oplus \mathcal{H}$ is positive (semidefinite) iff $||T||\leq 1$. My question is whether the usual method in the case of $M_2(\mathbb{C})$ where we can find that the eigen values for the matrix $\begin{bmatrix} 1 & b \\ b^{*} & 1\\ \end{bmatrix}$ are $|b|+1$ and $-|b|+1$, (implying positivity if $|b|\leq 1$) can be adapted to the non commutative setting. In other words, can we find a block form for the operator as $\begin{bmatrix} 1+\sqrt{T^*T} & 0 \\ 0 & 1-\sqrt{T^*T}\\ \end{bmatrix}$ and obtain the proof in this manner? If this won't work, any hints appreciated

Answer 1

Here is a hint: $$\begin{pmatrix}I&-T\\0&I\end{pmatrix}\begin{pmatrix}I&T\\T^*&I\end{pmatrix}\begin{pmatrix}I&0\\-T^*&I\end{pmatrix}=\begin{pmatrix}I-TT^*&0\\T^*&I\end{pmatrix}\begin{pmatrix}I&0\\-T^*&I\end{pmatrix}=\begin{pmatrix}I-TT^*&0\\0&I\end{pmatrix}.$$