Square Roots of Positive Bounded Operators on a Complex Hilbert Space, Commuting with Everything that Commutes with the Operator

In the setting of Complex Hilbert Spaces and Bounded Linear Maps: Standing Notation, let $H$ be a complex Hilbert space and let $T\in\mathcal{L}(H)$ with $T\ge0$. Then there is $S\in\mathcal{L}(H)$ such that

1. (Square root)¶ $S\ge0$ and $SS=T$; and

2. (Commutation)¶ $S$ commutes with every $B\in\mathcal{L}(H)$ that commutes with $T$.