Norm of $(T^*T)^\frac{1}{2}$

Question

Suppose that $T$ is an invertible linear map on a Hilbert space $H$ with $\|T\|=a$ and $\|T^{-1}\| = b$, then can we find a lower bound of $\|(T^*T)^\frac{1}{2}\|$ in terms of $a$ and $b$?

I would like to request for a hint. Thanks.

Answer 1

Since $$\lVert T^*T\rVert=\lVert TT^*\rVert=\lVert T\rVert^2$$ We have $$\|\sqrt{T^*T}\|=\sqrt{\|\sqrt{T^*T} (\sqrt{T^*T})^*\|}\\=\sqrt{\|\sqrt{T^*T}\sqrt{T^*T}\|}=\sqrt{\| T^*T\|}=\|T\|=a$$