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Keywords:
Hilbert space; linear operator; eigenvalue; Kato theorem; Weyl inequality
Hilbert space; linear operator; eigenvalue; Kato theorem; Weyl inequality
Summary:
Let $A$ be a bounded linear operator in a complex separable Hilbert space $\mathcal {H}$, and $S$ be a selfadjoint operator in $\mathcal {H}$. Assuming that $A-S$ belongs to the Schatten-von Neumann ideal $\mathcal {S}_p$ $(p> 1),$ we derive a bound for $\sum _{k}| {\rm R} \lambda _k(A)-\lambda _k(S)|^p$, where $\lambda _k(A)$ $(k=1, 2, \dots )$ are the eigenvalues of $A$. Our results are formulated in terms of the ``extended'' eigenvalue sets in the sense introduced by T. Kato. In addition, in the case $p=2$ we refine the Weyl inequality between the real parts of the eigenvalues of $A$ and the eigenvalues of its Hermitian component.
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