Inequality on Hilbert spaces

Question

Let $H$ be a complex Hilbert space. Is it possible that under certain conditions $$|\langle Tx,y\rangle|\leq \|T\||\langle x,y\rangle|$$ for all $x,y\in H$ with $x\not\perp y$ and $T\in B(H)$? Or actually, is this true for all Hilbert space $H$ and bounded linear operator $T$ on $H$? I couldn't find a counterexample.

Thanks for your ideas!

Answer 1

If the inequality holds whenever $ \langle x, y \rangle \neq 0$ it will hold when $ \langle x, y \rangle = 0$ also: Replace $y$ by $y+\frac 1n x$ and take limit as $n \to \infty$. Hence the comment by copper.hat provides a counter-example.