Exploring the spectra of some classes of paired singular integral operators: the scalar and matrix cases. (English) Zbl 1342.47006

The paper presents some results for finding the spectra of singular integral operators with essentially bounded matrix coefficients on the \(L^{2}\)-space on the unit circle (the standard singular integral operators of the form \(aI+bS\) and \(aI+SbI\) are called by the authors the paired singular integral operators). Spectral algorithms using symbolic and numerical computation with the help of the computer system Mathematica are discussed. Singular integral operators with rational coefficients and operators in the case of essentially bounded coefficients of some special class are investigated. The paper contains numerous examples.


47A10 Spectrum, resolvent
65R99 Numerical methods for integral equations, integral transforms
45E99 Singular integral equations
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)