Let $H$ be a Hilbert space with inner product $\langle\cdot,\cdot\rangle$. Fix a bounded operator $T$ on $H$, and $1\leq p<\infty$ (you can assume $p$ is an integer if necessary). Consider the following maps:
$$H\to[0,\infty):x\mapsto|\langle Tx,x\rangle|^p$$
and
$$H\to[0,\infty):x\mapsto\langle |T|^px,x\rangle.$$
Are these maps Lipschitz equivalent? That is, do there exist constants $c,C>0$ such that $$c|\langle Tx,x\rangle|^p\leq \langle |T|^px,x\rangle\leq C|\langle Tx,x\rangle|^p$$ for all $x\in H$? If not, are there any such inequalities between these quantities?