Comparing $R(A)$ and $R(A^{1/2})$ for positive operators

Question

Can you give me a bounded postive (self-adjoint) operator $A$ on a Hilbert space such that $R(A)\not\subset R(A^{1/2})$ where $R(A)$ designates the range of a linear mapping? Is this possible?

Thanks a lot!

Math.

Answer 1

It is impossible: $A = A^{1/2}\cdot A^{1/2}$, so everything in the range of $A$ is automatically in the range of $A^{1/2}$.

Answer 2

There is no such example. Note that for any operators $A,B$, we have $R(AB) \subset A$. Thus, we have $R(A) = R(A^{1/2}A^{1/2}) \subset R(A^{1/2})$.