Let $A$ be a positive operator on a real Hilbert space $\mathcal H.$ Is it always true that $\|A\| =\sup\limits_{\|x\| = 1} \left \langle Ax, x \right \rangle$?
I know that for self-adjoint operators the operator norm is given by the above expression. If the underlying Hilbert space is complex then I know that every positive operator is self-adjoint. But for real Hilbert spaces it is no longer true. So can we conclude the same (about operator norm) for real Hilbert spaces?
Any suggestion will be greatly appreciated. Thanks!
Best Answer
Take $A=\pmatrix{1 & n \\ -n & 1}$ then $\langle Ax,x\rangle = \|x\|_2^2$, but $\|A\|$ clearly depends on $n$.