Matrix norms translating with identity matrix

Question

Let $\mathcal{A}$ denote a linear operator, $\mathcal{A}^*$ its adjoint, and $\text{I}$ the identity matrix. Is the norm of $\mathcal{A}$ related to the norm of $\mathcal{A}^* \mathcal{A} - \text{I}$?

In particular, is it true that if

$\|\mathcal{A}^* \mathcal{A} - \text{I}\| \le 1/4$

then

$\|\mathcal{A}^* \mathcal{A}\| \ge 3/4$? If it is, why?

We know that $\|\mathcal{A}\|^2= \|\mathcal{A}^* \mathcal{A}\|$. I attempted proving it separating the norm of the difference with the triangle inequality.

Answer 1

Yes. The operator norm satisfies the triangle inequality, in particular: $$\Vert \mathrm I \Vert \leq \Vert \mathcal A^* \mathcal A \Vert + \Vert \mathrm I - \mathcal A^* \mathcal A \Vert \,,$$ so that $$\Vert \mathcal A^* \mathcal A \Vert \geq 1 - \Vert \mathcal A^* \mathcal A - \mathrm I\Vert \,.$$