Two classes of operators related to the perturbation classes problem

Manuel González1, Margot Salas-Brown2
1Departamento de Matemáticas, Facultad de Ciencias, Universidad de Cantabria, 39071, Santander, Spain
2Escuela de Ciencias Exactas e Ingeniería, Universidad Sergio Arboleda, Bogotá, Colombia

Tóm tắt

AbstractLet $${{\mathcal {S}}}{{\mathcal {S}}}$$ S S and $${{\mathcal {S}}}{{\mathcal {C}}}$$ S C be the strictly singular and the strictly cosingular operators acting between Banach spaces, and let $$P\Phi _+$$ P Φ + and $$P\Phi _+$$ P Φ + be the perturbation classes for the upper and the lower semi-Fredholm operators. We study two classes of operators $$\Phi {\mathcal {S}}$$ Φ S and $$\Phi {\mathcal {C}}$$ Φ C that satisfy $${{\mathcal {S}}}{{\mathcal {S}}}\subset \Phi {\mathcal {S}}\subset P\Phi _+$$ S S Φ S P Φ + and $${{\mathcal {S}}}{{\mathcal {C}}}\subset \Phi {\mathcal {C}}\subset P\Phi _-.$$ S C Φ C P Φ - . We give some conditions under which these inclusions become equalities, from which we derive some positive solutions to the perturbation classes problem for semi-Fredholm operators.

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