The operator norm of a mapping $\mathcal{A}:\mathbb{R}^{n_1\times n_2} \to \mathbb{R}^{n_1\times n_2}$ is defined by $$ \|\mathcal{A}\|=\sup_{\|X\|_{F} \leq 1}\|\mathcal{A}(X)\|_{F}, $$ where $X$ is a matrice, and $\|\cdot\|_{F}$ is $F$-norm defined by $\|X\|=\left<X,X\right>^{\frac{1}{2}}$.
If we have $\|\mathcal{A}\| \leq \kappa$, where $\kappa > 0$ is a constant, then we can get $\|\mathcal{A}(X)\|_{F}\leq \kappa \|X\|_{F}$ by the definition of operator norm. Can we find a constant $c >0$ such that $\|\mathcal{A}(X)\|_{F}\geq c \|X\|_{F}$. If not, can we add additional conditions to find thid $c$?