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We establish several inequalities for the spectral radius of a positive commutator of positive operators in a Banach space ordered by a normal and generating cone. The main purpose of this paper is to show that in order to prove the quasi-nilpotency of the commutator we do not have to impose any compactness condition on the operators under consideration. In this way we give a partial answer to the open problem posed in the paper by J. Bračič, R. Drnovšek, Y. B. Farforovskaya, E. L. Rabkin, J. Zemánek...
In the paper [13] we proved a fixed point theorem for an operator , which satisfies a generalized Lipschitz condition with respect to a linear bounded operator , that is:
The purpose of this paper is to show that the results obtained in [13], [14] can be extended to a nonlinear operator .
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