Generalized Rayleigh-Quotient Formulas for the Real Parts, Imaginary Parts, and Moduli of Simple Eigenvalues of Compact Operators

Year 2022, Volume: 5 Issue: 2, 48 - 77, 30.06.2022

Ludwig Kohaupt

https://doi.org/10.33434/cams.1020515

Abstract

In an earlier paper, the author derived generalized Rayleigh-quotient formulas for
the real parts, imaginary parts, and moduli of the eigenvalues of diagonalizable
matrices. More precisely, max-, min-max-, min-, and max-min-formulas were
obtained. In this paper, we extend these results to the eigenvalues of linear
nonsymmetric compact operators with simple eigenvalues in a Hilbert space. As
an application, a new formula for the spectral radius is derived. An example
arising from a boundary value problem in Mathematical Physics illustrates the
general results, and numerical computations underpin the theoretical findings.
In addition, the Euler column is treated from the area of Elastomechanics, which
is complemented by references to other examples from this area.

Keywords

Generalized Rayleigh-Quotients, Hilbert space, Real parts, Imaginary parts, and moduli of eigenvalues, Simple eigenvalues of compact operators

Supporting Institution

There is no supporting institution

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