Let $H$ be a $\mathbb R$-Hilbert space and $A\in\mathfrak L(H)$ be self-adjoint with $$\left\|Ax\right\|_H\le\left\|x\right\|_H\;\;\;\text{for all }x\in H.\tag1$$
Let $K\subseteq H$ be closed and suppose I want to show that $$\left\|Ax\right\|_H\le c\left\|x\right\|_H\;\;\;\text{for all }x\in K\tag2$$ for some $c\in[0,1)$. Now assume that I'm able to minimize $\sup_{x\in K}\langle Ax,x\rangle_H$. Are we able to relate the obtained bound to $c$?