# Operator Norm of a Self-adjoint Operator

I am trying to prove $$\|A\|=\sup_{||x||\leq 1}|\langle x,Ax\rangle|$$ for a self adjoint bounded linear operator $$A$$ on a complex Hilbert space. I know how to prove $$\|A\|\geq\sup_{||x||\leq 1}|\langle x,Ax\rangle|$$. For the converse, however, $$\|A\| = \sup_{||x||\leq 1} \| Ax \|$$ $$= \sup_{||x||\leq 1} \langle Ax,Ax\rangle^{1/2}$$ $$= \sup_{||x||\leq 1} \langle x,A^*Ax\rangle^{1/2}$$ $$= \sup_{||x||\leq 1} \langle x,A^2x\rangle^{1/2}$$ And then I'm stuck!

One can easily verify that $$\|A\|=\sup\{|\langle Ax,y\rangle| : x,y \in \mathcal H,\ \|x\|=\|y\|=1\}$$ in a complex Hilbert space $$\mathcal H$$.
Notice that $$\langle A(x+y), x+y\rangle − \langle A(x- y), x- y\rangle = 2\langle Ax,y\rangle+2\langle Ay,x\rangle.$$ Since $$A$$ is self adjoint, $$\langle Ay,x\rangle=\langle y,Ax\rangle=\overline{\langle Ax,y\rangle}.$$ So $$\langle A(x+y), x+y\rangle − \langle A(x- y), x- y\rangle=4\newcommand{\re}{\operatorname{Re}}\re\langle Ax,y\rangle.$$ Let $$P:=\sup_{||x||\leq 1}|\langle x,Ax\rangle|.$$ Then \begin{align*} |4\re\langle Ax,y\rangle| &=|\langle A(x+y), x+y\rangle − \langle A(x- y), x- y\rangle |\\ &\leq P\|x+y\|^2+P\|x-y\|^2\\ &=2P(\|x\|^2+\|y\|^2). \end{align*} So, whenever $$\|x\|=\|y\|=1$$, we have $$|\re\langle Ax,y\rangle|\leq P\tag{\color{red}{1}}\label1.$$
Suppose $$\langle Ax,y\rangle=re^{i\theta}$$ with $$\|x\|=\|y\|=1$$. I will construct an element $$z$$ with $$\|z\|=1$$ such that $$|\langle Ax,y\rangle|=|\re\langle Az,y\rangle|$$. Then we can apply $$(\ref 1)$$ to $$|\re\langle Az,y\rangle|$$ and we will get $$|\langle Ax,y\rangle|\leq P$$.
Consider $$z=e^{-i\theta} x$$. Then $$\|z\|=1$$. Also, note that $$\langle Az,y\rangle=e^{-i\theta}\langle Ax,y\rangle=r=\re\langle Az,y\rangle,$$ and $$|\langle Ax,y\rangle|=r$$. So $$|\langle Ax,y\rangle|=r=|\re\langle Az,y\rangle|\leq P.$$ Hence $$\|A\|\leq P$$.
• Can you please explain how you got $|\langle A(x+y), x+y\rangle − \langle A(x- y), x- y\rangle |\leq P\|x+y\|^2+P\|x-y\|^2$? Nov 17, 2019 at 20:42