Equality of expressions containing supremum of some double sums

Question

I am doing a project related to operator norm and some double sequences. In the course of proving some results, I encounter the following expressions:

$\displaystyle \|\alpha\| := \sup_{\|x\|_p=1} \left[\sum_{j=1}^\infty \left|\sum_{i=1}^\infty x_i \alpha_{ij}\right|^q\right]^{\frac{1}{q}}$

and

$\displaystyle \|\alpha\|^* := \left(\sum_{j=1}^\infty \sum_{i=1}^\infty |\alpha_{ij}|^q \right)^{\frac{1}{q}}$,

where $\|\cdot\|_p$ is the $\ell_p$ norm, $p$ is the conjugate of $q$, and $x=(x_i)$ is a sequence, $\alpha=(\alpha_{ij})$ is a double-indexed sequence.

I want to say that they are equal. But is it true that $\|\alpha\|= \|\alpha\|^*$ ?

Thank you.

Answer 1

You can take $\alpha_{11}=\alpha_{22}=1$, and all other coefficients equal to $0$. Let us fix $p=q=2$. Then $$ \|\alpha\|^*=2^{1/2}, $$ while, for $x$ with $\|x\|_2=1$, $$ \left[\sum_{j=1}^\infty \left|\sum_{i=1}^\infty x_i \alpha_{ij}\right|^2\right]^{\frac{1}{2}}=(|x_1|^2+|x_2|^2)^{1/2}\leq\|x\|_2=1. $$ So $\|\alpha\|=1$.