Geodesic
Perturbations of real parts of eigenvalues of bounded linear operators in a Hilbert space
Czechoslovak Mathematical Journal, Tome 74 (2024) no. 2, pp. 567-573
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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.
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.
DOI : 10.21136/CMJ.2024.0468-23
Classification : 47A10, 47A55, 47B10
Keywords: Hilbert space; linear operator; eigenvalue; Kato theorem; Weyl inequality 
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     title = {Perturbations of real parts of eigenvalues of bounded linear operators in a {Hilbert} space},
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Gil', Michael. Perturbations of real parts of eigenvalues of bounded linear operators in a Hilbert space. Czechoslovak Mathematical Journal, Tome 74 (2024) no. 2, pp. 567-573. doi: 10.21136/CMJ.2024.0468-23

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