If $T$ is a positive operator, and with $S^2=T$, then $S$ must be self-adjoint.

Question

Let $S$ and $T$ be a linear operator on finite dimensional vector space $V$. If $T$ is a positive operator, and with $S^2=T$, then $S$ must be self-adjoint. Does this statement hold?

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I try to use the polar decomposition of operators, there exists $U$ unitary, $R$ positive so that $T=UR$.

I found a similar question: Show there exists $S$ which is self-adjoint such that $S^2=T^*T$ and $S$ is invertible.

Answer 1

The statement is not true. Let $$S=\begin{pmatrix} 1 & 1\\ 0 & -1\end{pmatrix}$$ Then $$S^2=\begin{pmatrix} 1& 0\\ 0 &1\end{pmatrix}$$ but $S$ is not self-adjoint.