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An application of Plemelj-Smithies formulas to computing generalized inverses of Fredholm operators

Ciecierska G.

Journal of Applied Mathematics and Computational Mechanics

2015

Vol. 14, nr 1

13--26

We consider continuous operators S +T in Banach spaces, where S is Fredholm and T is quasinuclear. By referring to the basic result of the Fredholm theory, i.e. to the expression of the resolvent ( I + λT ) −1 of the operator T as a quotient of entire functions of λ , we derive analogous formulas for generalized inverses of operators S +T. We apply the Plemelj-Smithies formulas describing terms of determinant systems for the quasinuclear perturbations of Fredholm operators.

Throught a generalized inverse

Stępniak C.

Demonstratio Mathematica

2008

Vol. 41, nr 2

291-296

Traditionally, the existence of a generalized inverse of a matrix A is derived in an indirect way from the matrix equation AXA = A. We reach this result in a direct and constructive manner, based on spectral decomposition. Moreover, some new results on its characterization and on representation of the entire set of generalized inverses are given. Usefulness of these results is demonstrated in examples.