Inequalities for imaginary parts of eigenvalues of Schatten–von Neumann operators

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Let Sp be the p-Schatten–von Neumann ideal of operators in a Hilbert space, A∈ Sp(2 ≤ p< ∞) , AR= (A+ A) / 2 , AI= (A- A) / (2 i) and λk(A) (k= 1 , 2 ,..) are the eigenvalues of A. The asterisk means the adjointness. We derive the inequality ∑k=1∞|Imλk(A)-λk(AI)|p≤ηpNpp(AR), where ηp≤ 2 p is an explicitly defined constant dependent on p, only, and Np(.) is the norm of Sp . In addition, assuming that AI is a nuclear operator, we obtain an inequality for the series of the imaginary parts of the eigenvalues of A via the trace of AI .

Original languageEnglish
Pages (from-to)801-807
Number of pages7
JournalRendiconti del Circolo Matematico di Palermo
Issue number3
StateAccepted/In press - 1 Jan 2023


  • Compact operators
  • Hilbert space
  • Localization of eigenvalues
  • Perturbations

ASJC Scopus subject areas

  • General Mathematics


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