Is there a condition for $\| T\| =\| T^\ast\|$ where $T$ is a linear operator.

Question

I will first motivate my question:

For both the Frobenius norm and the operator norm, we can show that for a linear operator $T$, \begin{align} \| T \|_F^2 = Tr( T T^\ast) = Tr(T^\ast T) = \| T^\ast \|_F^2 \end{align} Same chain of equalities can be repeated with the operator norm (the induced $\ell_2$ norm).

Is it a general property of norms? Is there a condition for which norms we can write $\| T\| =\| T^\ast\|$.

Answer 1

Let $T : H \longrightarrow H$ be a bounded (continuous) linear map from a Hilbert space $H$ to itself. Then under the following definition of the operator norm $\|T\| = \sup\{|(Tf, g)| : \|f\|\leq1, \|g\|\leq1 \}$ we have that $\|T\| = \|T^*\|$. Indeed,

$\|T\| = \sup\{|(Tf, g)| : \|f\|\leq1, \|g\|\leq1 \} = \sup\{|(f, T^*g)| : \|f\|\leq1, \|g\|\leq1 \} = \|T^*\|$.

This was taken from chapter 5.2 of Stein and Shakarchi's Book 3 on Real Analysis (Page 183).