Symbolic computation applied to the study of the kernel of special classes of paired singular integral operators - Conceição, Ana C. (2021)
Operator theory has many applications in several main scientific research areas (structural mechan-ics, aeronautics, quantum mechanics, ecology, probability theory, electrical engineering, among others) and theimportance of its study is globally acknowledged. On the study of the operator’s kernel some progress has beenachieved for some specific classes of singular integral operators whose properties allow the use of particular strate-gies. However, the existing algorithms allow, in general, to study the dimension of the kernel of some classes ofsingular integral operators but are not designed to be implemented on a computer. The main goal of this paper is toshow how the symbolic and numeric capabilities of a computer algebra system can be used to study the kernel ofspecial classes of paired singular integral operators with essentially bounded coefficients defined on the unit circle.It is described how some factorization algorithms can be used to compute the dimension of the kernel of specialclasses of singular integral operators. The analytical algorithms [ADimKerPaired-Scalar], [AKerPaired-Scalar],and [ADimKerPaired-Matrix] are presented. The design of these new algorithms was focused on the possibility ofimplementing on a computer all the extensive symbolic and numeric calculations present in the algorithms. Forthe essentially bounded hermitian coefficients case, there exist some relations with Hankel operators. The papercontains some interesting and nontrivial examples obtained with the use of a computer algebra system.