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JOURNALS // Sibirskii Matematicheskii Zhurnal // Archive

Sibirsk. Mat. Zh., 2024 Volume 65, Number 1, Pages 87–91 (Mi smj7842)

Hilbert–Pólya operators in Krein spaces

V. V. Kapustin

St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences

Abstract: We construct some class of selfadjoint operators in the Krein spaces consisting of functions on the straight line $\{\operatorname{Re}s=\frac12\}$. Each of these operators is a rank-one perturbation of a selfadjoint operator in the corresponding Hilbert space and has eigenvalues complex numbers of the form $\frac1{s(1-s)}$, where $s$ ranges over the set of nontrivial zeros of the Riemann zeta-function.

Keywords: Riemann zeta-function, eigenvalue, perturbation, selfadjoint operator.

UDC: 517.984

MSC: 35R30

Received: 29.11.2022
Revised: 29.11.2022
Accepted: 28.11.2023

DOI: 10.33048/smzh.2024.65.108



© Steklov Math. Inst. of RAS, 2025