Relationship between eigenvalues of compact operators $A$ and $(A+A^*)/2$

Question

A result from 'Topics in Matrix Analysis' by Horn & Johnson (3.3.33) is the following: For $A\in \mathbb{M}_n$, $\sum_{i=1}^k Re \lambda_i(A) \leq \sum_{i=1}^k Re \lambda_i \big(\frac{A+A^*}{2}\big), ~ k =1,...,n,$ with equality for $k=n.$ I want to know if this result holds true for non-positive, non self-adjoint compact operators (on a Hilbert space). In this case, all we know is that the eigenvalues have $0$ as an accumulation point, the summations above might not even be finite for $k=\infty$ case. I have been able to show this is true in the case of the largest eigenvalues, would an induction type argument work for $k=n$? Any help would be appreciated.