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Spectral radius inequalities for positive commutators

Mirosława Zima — 2014

Czechoslovak Mathematical Journal

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...

Applications of the spectral radius to some integral equations

Mirosława Zima — 1995

Commentationes Mathematicae Universitatis Carolinae

In the paper [13] we proved a fixed point theorem for an operator 𝒜 , which satisfies a generalized Lipschitz condition with respect to a linear bounded operator A , that is: m ( 𝒜 x - 𝒜 y ) ≺ A m ( x - y ) . The purpose of this paper is to show that the results obtained in [13], [14] can be extended to a nonlinear operator A .

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