Let $F_1,F_2$ be two complex Hilbert spaces. Consider \begin{equation*} T=\begin{pmatrix}A & B \\ C & D \end{pmatrix}\in \mathcal{B}(F_1\oplus F_2). \end{equation*}
If $T$ is a positive operator on $F_1\oplus F_2$. Is the square root of $T$ given by $$T^{1/2}=\begin{pmatrix}A^{1/2} & B^{1/2} \\ C^{1/2} & D^{1/2} \end{pmatrix}?$$
Besides the fact that $B$ and $C$ need not be positive (not even selfadjoint), what you ask is not true even if $F_1$ and $F_2$ are one-dimensional. Let $T\in M_2(\mathbb C)$ be $$ T=\begin{bmatrix} 1&1\\ 1&1\end{bmatrix}. $$ Then according to your formula you would expect to have $T^{1/2}=T$, but $T^2=2T$, so $T^{1/2}=\frac1{\sqrt2}\,T$. You also have that $$ \begin{bmatrix} 1&-i\\i&1\end{bmatrix}\geq0, $$ and it's not even clear how to apply your formula.
For a more dramatic example, let $$ T=\begin{bmatrix} 1&1\\1&4\end{bmatrix} . $$ Then its unique positive square root is $$ T^{1/2}=\frac{\sqrt{5-2\sqrt3}}{\sqrt{13}}\,\begin{bmatrix} 1+\sqrt3&1\\ 1& 4+\sqrt3 \end{bmatrix} $$