Let $A$ be a positive self-adjoint operator on a Hilbert space $\mathcal H.$ If $\inf\limits_{\|x\|=1} \left \langle Ax, x \right \rangle = 0$ then $A$ cannot be invertible.
How do I prove it? Any help will be greatly appreciated.
Thanks for giving time.
If $A$ were invertible, then $A^{1/2}$ would also be invertible (with inverse $A^{-1/2}$). In particular, $A^{1/2}$ would be bounded below - there would exist $\epsilon > 0$ such that $$ \|A^{1/2}x\| \geq \epsilon\|x\|. $$ Squaring both sides and writing out the inner product gives $\langle Ax,x\rangle$ on the left-hand-side. This should give you the contradiction you seek.