Let $A,B$ be two Trace class operators with spectral decomposition $\sum_{j\geq 1} \lambda_j \phi_j(\cdot)\otimes \phi_j(\cdot)$ and $\sum_{j\geq 1} \gamma_j \psi_j(\cdot)\otimes \psi_j(\cdot)$ respectively. Can one say anything about the eigenvalues and eigenvectors of $A^{1/2}BA^{1/2}$ maybe under further assumptions ? In particular I am curious if clean relationships exist between the eigenvalues of $A^{1/2}BA^{1/2}$ and $\lambda_j$, $\gamma_j$, for $j\geq 1$ in some special cases ?
3
-
1$\begingroup$ the eigenvalues of $A^{1/2}BA^{1/2}$ are the same those of $AB$. $\endgroup$– Carlo BeenakkerCommented Aug 4, 2020 at 6:25
-
$\begingroup$ Are you assuming $A$ is a positive trace class operator? Otherwise $A^{1/2}$ doesn't make sense to me.. $\endgroup$– Darth VaderCommented Aug 4, 2020 at 23:46
-
$\begingroup$ @Darth Vader Yes indeed $\endgroup$– KcafeCommented Aug 5, 2020 at 1:47
Add a comment
|