Minimal Normalization of Wiener–Hopf Operators in Spaces of Bessel Potentials

AbstractA class of operators is investigated which results from certain boundary and transmission problems, the so-called Sommerfeld diffraction problems. In various cases these are of normal type but not normally solvable, and the problem is how to normalize the operators in a physically relevant way, i.e., not loosing the Hilbert space structure of function spaces defined by a locally finite energy norm. The present approach solves this question rigorously for the case where the lifted Fourier symbol matrix function is Hölder continuous on the real line with a jump at infinity. It incorporates the intuitive concept of compatibility conditions which is known from some canonical problems. Further it presents explicit analytical formulas for generalized inverses of the normalized operators in terms of matrix factorization

Visualizing Rank Deficient Models: A Row Equation Geometry of Rank Deficient Matrices and Constrained-Regression

Situations often arise in which the matrix of independent variables is not of full column rank. That is, there are one or more linear dependencies among the independent variables. This paper covers in detail the situation in which the rank is one less than full column rank and extends this coverage to include cases of even greater rank deficiency. The emphasis is on the row geometry of the solutions based on the normal equations. The author shows geometrically how constrained-regression/generalized-inverses work in this situation to provide a solution in the face of rank deficiency

The weak core inverse

[EN] In this paper, we introduce a new generalized inverse, called weak core inverse (or, in short, WC inverse) of a complex square matrix. This new inverse extends the notion of the core inverse defined by Baksalary and Trenkler (Linear Multilinear Algebra 58(6):681-697, 2010). We investigate characterizations, representations, and properties for this generalized inverse. In addition, we introduce weak core matrices (or, in short, WC matrices) and we show that these matrices form a more general class than that given by the known weak group matrices, recently investigated by H. Wang and X. Liu.In what follows, we detail the acknowledgements. D.E. Ferreyra, F.E. Levis, A.N. Priori - Partially supported by Universidad Nacional de Rio Cuarto (Grant PPI 18/C559) and CONICET (Grant PIP 112-201501-00433CO). D.E. Ferreyra F.E. Levis - Partially supported by ANPCyT (Grant PICT 201803492). D.E. Ferreyra, N. Thome -Partially supported by Universidad Nacional de La Pampa, Facultad de Ingenieria (Grant Resol. Nro. 135/19). N. Thome -Partially supported by Ministerio de Economia, Industria y Competitividad of Spain (Grant Red de Excelencia MTM2017-90682-REDT) and by Universidad Nacional del Sur of Argentina (Grant 24/L108). We would like to thank the Referees for their valuable comments and suggestions which helped us to considerably improve the presentation of the paperFerreyra, DE.; Levis, FE.; Priori, AN.; Thome, N. (2021). The weak core inverse. Aequationes Mathematicae. 95(2):351-373. https://doi.org/10.1007/s00010-020-00752-zS351373952Ben-Israel, A., Greville, T.N.E.: Generalized Inverses: Theory and Applications, 2nd edn. Springer, New York (2003)Baksalary, O.M., Trenkler, G.: Core inverse of matrices. Linear Multilinear Algebra 58(6), 681–697 (2010)Baksalary, O.M., Trenkler, G.: On a generalized core inverse. Appl. Math. 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