Timeline for Extending Ky Fan's eigenvalues inequality to kernel operators
Current License: CC BY-SA 4.0
4 events
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| Aug 23, 2021 at 7:49 | comment | added | Giorgio Metafune | Yes, certainly. Assume that the formula with "max" holds. Then apply it to $A+B$ and estimate the max for $A+B$ with the sum of the maxima. You find the formula with max in the original paper by Ky-Fan (On inequalities by Weyl, I) for matrices, but the same fomula holds for compact self-adjoint operators, using the spectral theorem to diagonalyse. Hope now is more clear, if not do not esitate to write again. | |
| Aug 23, 2021 at 7:21 | comment | added | user43389 | @Giorgio, Could you maybe expand on why that's the case? | |
| Aug 22, 2021 at 23:16 | comment | added | Giorgio Metafune | If you use that $\sum_{i=1}^k \lambda_i(A)= \max \sum_{i=1}^k (Ax_i,x_i)$, where the maximum is taken over all orthonormnal sets $\{x_1, \dots,x_k\}$, then the inequalities hold whenever $A,B$ are compact self-adjoint operators in a Hilbert space. | |
| Aug 22, 2021 at 20:20 | history | asked | user43389 | CC BY-SA 4.0 |