Abstract
Firstly, we study the stability of the Moore-Penrose invertibility of Hilbert space operators, establishing necessary and sufficient conditions for the invariance of the Moore-Penrose invertibility under small perturbations, compact perturbations and small compact perturbations. Secondly, we study the lifting problem of Moore-Penrose invertible elements of the Calkin algebra, showing that essentially Moore-Penrose invertible operators are small compact perturbations of Moore-Penrose invertible operators. Finally, we discuss the compactness of Moore-Penrose spectra, showing that each Hilbert space operator can be perturbed into an operator with nonempty compact Moore-Penrose spectrum by an arbitrarily small compact perturbation.
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