Norm of operator $A$ st. $A^2 = I$?

Question

I'm wondering what can be said about the norm $||A||$ of an operator which squares to identity. All I can think of is that

$$1=||AA|| \leq ||A||^2$$

so that $||A|| \geq 1$. But can anything else be said? I'd really like it to be always equal to $1$ for a problem I'm working on, but can't seem to prove it.

The norm I'm considering is this: $$||A|| = \sup_{x \in H} \frac{|Ax|}{|x|}$$

$H$ is a Hilbert space.

Answer 1

If $A$ is symmetric (Hermitian), then that's true (assuming you're talking about the induced Euclidean norm). However, in general, $\|A\|$ can be very large. For example, take $$ A=\pmatrix{1&t\\0&-1} $$ with $t$ as large as you want to make it.

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Note that for any operator on a Hilbert space, we have $\|A^*A\| = \|A\|^2$ (you should have this as a theorem somewhere). Thus, we have $$ \|A\|^2 = \|A^*A\| = \|A^2\| = \|I\| = 1 $$ so indeed, $\|A\| = 1$.

Answer 2

It can be anything greater than $1$, take for instance $$ A=\begin{bmatrix}0&x\\1/x&0\end{bmatrix} $$