# Finding positive semi-definite counterexamples for self-adjoint bounded operators on a Hilbert Space.

Let $$\mathcal{H}$$ be a Hilbert space. Assume $$x, y \in B(\mathcal{H})$$ are self-adjoint. We say $$x \geq y$$ if $$x-y$$ is positive semi-definite. We define positive semi-definite as $$\langle x \xi, \xi \rangle \geq 0$$ for all $$\xi \in \mathcal{H}$$.

I am looking for specific examples to illustrate the following statements:

(1) $$x,y \geq 0 \nRightarrow xy \geq 0.$$

I looked for an example of two matrices such that both are positive semi-definite but their product is not, but I couldn't find an example where the matrices are symmetric (since I want $$x,y$$ self-adjoint). I only saw examples online where the matrices were not symmetric.

(2) $$x \geq y \geq 0 \nRightarrow x^2 \geq y^2.$$

By the definition above, we are looking for $$x$$ and $$y$$ self-adjoint such that $$(x-y)$$ and $$y$$ are both positive semi-definite, but $$(x^2-y^2)$$ is not. Again I looked for matrices that would satisfy this but had no luck.

Are matrices over $$\mathbb{C}$$ a bad example of $$B(\mathcal{H})$$ to take for these situations?

For the first one: consider $$A = \begin{pmatrix} 2 & 1 \\ 1 & 1\end{pmatrix} \text{ and } B = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}.$$ Both are positive, but $$AB = \begin{pmatrix} 2 & 0 \\ 1 & 0 \end{pmatrix},$$ which is not self-adjoint, hence not positive.
For the second one, lets use the same matrices! Clearly $$A \geq B \geq 0$$ (as $$A - B$$ is a positive multiple of a projection), but $$A^2 - B^2 = \begin{pmatrix} 4 & 3 \\ 3 & 2 \end{pmatrix},$$ which has eigenvalue $$3 - \sqrt{10} < 0$$, so it cannot be positive.
• Positive semi-definite here requires being self-adjoint (symmetric when the entries are real). An operator $T$ is positive (i.e., $\langle T\xi,\xi \rangle \geq 0 \forall \xi$) if and only if $T$ is self-adjoint and the spectrum of $T$ is is contained in $[0,\infty)$. In $M_2$, the spectrum is just the set of eigenvalues, so having non-negative eigenvalues is not enough to ensure positivity of $AB$. Commented Aug 31, 2020 at 16:56