Matrix operator $p$-norm and eigenvalues Denote by $\|\cdot\|_p$ the operator $p$-norm for matrices in $\mathcal M_{d\times d}(\mathbb R)$.
Let $a_n \to 0$ be a sequence of positive real numbers, and $\mathbf A$ a positive definite matrix.
We know that when $p=2$, $\|\mathbf I-a_n \mathbf A \|_2 = 1-a_n \lambda_{\min}(\mathbf A)$ when $n$ is large.
Question: When $1<p<2$ or $p>2$, is there a constant $K$, such that
$$  \|\mathbf I-a_n\mathbf A \|_p \leqslant 1-Ka_n $$
holds for large $n$?

I finally solved this problem in my paper.
Behold the preprint of it (see Section 3.1 onwards): https://arxiv.org/abs/2102.10346
