Square root of positive matrix operator 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 
$$
T^{1/2}=\frac{\sqrt{5-2\sqrt3}}{\sqrt{13}}\,\begin{bmatrix}
1+\sqrt3&1\\
1& 4+\sqrt3 
\end{bmatrix}
$$
