Suppose that $H$ is a Hilbert space and $A: H \rightarrow H$, $B: H\rightarrow H$ are self-adjoint linear operators such that $0\leq A \leq B$. If $B$ is compact, then it is bounded, but can it be proved that $A $ is bounded? Thanks a lot in advance!
2026-03-27 01:50:43.1774576243
If $0\leq A \leq B$, is operator $A$ bounded?
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The condition $\forall x,y\in H\quad \langle Ax,y\rangle = \langle x,Ay\rangle $ implies $A$ is bounded. E.g by Uniform Boundedness Principle. Details in the accepted answer: link