Let $T$ be a bounded operator. Then, the operators $\left\lvert T \right\rvert:=\sqrt{T^*T}$ and $\left\lvert T^* \right\rvert:=\sqrt{TT^*}$ are well-defined.

Is there a way to write $(\left\lvert T \right\rvert -i)^{-1} -(\left\lvert T^* \right\rvert -i)^{-1}$ in terms of operators that are more accessible from $T$ than the absolute values?-The problem is that taking the square root in operator theory is a very non-explicit process. Or is there a way to simplify this difference if the explicit structure of $T$ is well-known? Of course, I thought about the resolvent identity, but $$(\left\lvert T \right\rvert -i)^{-1} (\left\lvert T^* \right\rvert-\left\lvert T \right\rvert) (\left\lvert T^* \right\rvert -i)^{-1}$$ looks even more difficult.

Let $T$ be a bounded operator. Then, the operators $\left\lvert T \right\rvert:=\sqrt{T^*T}$ and $\left\lvert T^* \right\rvert:=\sqrt{TT^*}$ are well-defined.

Is there a way to write $$(\left\lvert T \right\rvert -i)^{-1} -(\left\lvert T^* \right\rvert -i)^{-1}$$ in terms of operators that are more accessible from $T$ than the absolute values?

The problem is that taking the square root in operator theory is a very non-explicit process. Or is there a way to simplify this difference if the explicit structure of $T$ is well-known? Of course, I thought about the resolvent identity, but $$(\left\lvert T \right\rvert -i)^{-1} (\left\lvert T^* \right\rvert-\left\lvert T \right\rvert) (\left\lvert T^* \right\rvert -i)^{-1}$$ looks even more difficult.