If $H$ is finite dimensional there is a one-line solution: ${\rm tr}(JK - KJ) = 0$, so $JK - KJ$ cannot have positive spectrum.

But it is false in general! Let $V$ be a partial isometry from $H$ onto a proper subspace $K$, so that $V^*V = I$ and $VV^* = P \neq I$ is the projection onto $K$. Then let $J = V + V^*$ and $K = V - V^*$; we have $[J,K] = 2(V^*V - VV^*) = 2(I - P) \geq 0$, but it is not zero.

If $H$ is finite dimensional there is a one-line solution: ${\rm tr}(JK - KJ) = 0$, so $JK - KJ$ cannot have positive spectrum.

But it is false in general! Let $V$ be a partial isometry from $H$ onto a proper subspace $H_0$, so that $V^*V = I$ and $VV^* = P \neq I$ is the projection onto $H_0$. Then let $J = V + V^*$ and $K = V - V^*$; we have $[J,K] = 2(V^*V - VV^*) = 2(I - P) \geq 0$, but it is not zero.

One-line solution: ${\rm tr}(JK - KJ) = 0$, so $JK - KJ$ cannot have positive spectrum.

If $H$ is finite dimensional there is a one-line solution: ${\rm tr}(JK - KJ) = 0$, so $JK - KJ$ cannot have positive spectrum.

But it is false in general! Let $V$ be a partial isometry from $H$ onto a proper subspace $K$, so that $V^*V = I$ and $VV^* = P \neq I$ is the projection onto $K$. Then let $J = V + V^*$ and $K = V - V^*$; we have $[J,K] = 2(V^*V - VV^*) = 2(I - P) \geq 0$, but it is not zero.