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

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