Given a Hilbert Space $H$ and a self-adjoint, positive operator $A$ on $H$.
Is it true that $\|A^{1/2}x\| \le \|Ax\|$ for all $x \in D(A)$?
If yes, how would the proof go?
2026-04-01 22:43:13.1775083393
Comparing an Operator and its Square Root
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It is not true. Consider the operator $\frac{1}{4} \cdot {1}_{H}$ (i.e. one fourth times the identity on $H$). It is clearly positive definite and the inequality is not true for any nonzero $x \in H$.