Inequality using Operator norm with differential of Matrix Function

Question

I'm trying to show that

Inequality $\,\| A^2 − B^2 \| ≤ 2\|A − B\|\,$ holds, where $\|\cdot\|$ is the Operator norm, and $A,B \in \mathbb R^{n,n}\,$ with $\|A\| < 1$ and $\| B\| < 1$.

I've shown that for $F(A)=A^{2}$ the differential is $\big(F'(A)\big)(H)=AH+HA$.

How can I show that $\|F'(X)\| \leq 2$ for $ X \in \{A+t(B-A)\mid t \in [0,1]\}$ ?

Answer 1

Note that $$\|F'(X)H\|=\|XH+HX\|\leq\|XH\|+\|HX\|\leq\|X\|\|H\|+\|H\|\|X\|=2\|X\|\|H\|$$ Furthermore for $X\in \{A+t(B-A)\mid t\in[0,1]\}$ we have $$\|X\|= \|(1-t)A+tB\|\leq\|(1-t)A\|+\|tB\|<(1-t)+t=1$$ Hence $\|F'(X)\|<2$.