The question $2$-norm of idempotent matrix was closed, but I didn't think the answer was obvious, although an answer has been provided via a link in a comment. I think that comment could be left as an answer, so I have voted to reopen.

It also seems to me that some people who voted to close, and some who left comments, interpreted $\Vert \cdot\Vert_2$ as meaning the Hilbert--Schmidt norm whereas I think it is meant to be the operator norm. (Considering diagonal $3\times 3$ matrices shows that the result is false for any Schatten p-norm with $p<\infty$.)

[REOPENED]

The question $2$-norm of idempotent matrix was closed, but I didn't think the answer was obvious, although an answer has been provided via a link in a comment. I think that comment could be left as an answer, so I have voted to reopen.

It also seems to me that some people who voted to close, and some who left comments, interpreted $\Vert \cdot\Vert_2$ as meaning the Hilbert--Schmidt norm whereas I think it is meant to be the operator norm. (Considering diagonal $3\times 3$ matrices shows that the result is false for any Schatten p-norm with $p<\infty$.)