I am wondering what are the best known relationship between $\|A\|_{2\rightarrow 2}$ and $\|A\|_{\infty\rightarrow 2}$ and how tight it is.

E.g., the trivial result is that for matrix $A$ with dimension $d\times d$

$$\|A\|_{2\rightarrow 2} \le \|A\|_{\infty\rightarrow 2}\le \sqrt{d} \|A\|_{2\rightarrow 2}$$

In particularly I am wondering whether the second inequality is tight, and if yes, do we have a good understanding of when it's close to be tight? (e.g., do we have any existing construction of $A$ so that it's tight?)

The same question can be asked for other induced norms as well, which I am also curious about.

Thanks!

I am wondering what are the best known relationship between $\|A\|_{2\rightarrow 2}$ and $\|A\|_{\infty\rightarrow 2}$ and how tight it is.

E.g., the trivial result is that for matrix $A$ with dimension $d\times d$

$$\|A\|_{2\rightarrow 2} \le \|A\|_{\infty\rightarrow 2}\le \sqrt{d} \|A\|_{2\rightarrow 2}$$

In particularly I am wondering whether the first inequality is tight (up to universal constant factors), and if yes, do we have a good understanding of when it's close to be tight? (e.g., do we have any existing construction of $A$ so that it's tight?)

(I was asking about the second inequality but I realized that I meant the first inequality)

The same question can be asked for other induced norms as well, which I am also curious about.

Thanks!