bounded and unital operator

Question

‎let‎ ‎$ T \in B ( H ) $‎be ‎invertible.‎

how ‎can ‎help ‎me ‎to ‎prove:‎

$ ‎\parallel ‎T‎ ‎\parallel =‎ ‎‎\parallel T‎‎‎^{‎-1‎} ‎‎‎‎‎\parallel‎‎‎ $ ‎if ‎only ‎if‎ ‎$ T $ ‎is ‎Unital?‎

‎$ T $ ‎is unital means ‎$ T ‎T‎^{*} = T‎^{*} T = I ‎$ ‎‎and ‎$‎B ( H ) ‎‎$‎means bounded operator on Hilbert space‎‎‎‎.

thanks ‎for ‎your ‎attention

Answer 1

Definitely not true: $$T=\begin{pmatrix}2&0\\0&\frac12\end{pmatrix}\qquad T^{-1}=\begin{pmatrix}\frac12&0\\0&2\end{pmatrix}$$ Clearly $\|T\|=\|T^{-1}\|$ but neither is unital.