Suppose we have a positive operator $A \in \mathcal{B}(\mathcal{H})$, does it follow that $$\|A\|^{1/2} = \|A^{1/2}\|?$$

If not, is there some relation between these quantities?

  • $\begingroup$ Where did this peoblem come from? $\endgroup$ – science Mar 5 '15 at 2:55
  • $\begingroup$ A colleague of mine was using the polar decomposition of a operator $T = U |T|$ where $|T| = (T^* T)^{1/2}$. To help him solve the problem he used this fact. I suspect its false, but I am busy with homework so I do not wish to think about it. $\endgroup$ – Ryan Mar 5 '15 at 2:57

Yes. Use the fact that by the spectral theorem, the norm of a positive operator is its spectral radius.

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