Comments on the cosmic convergence of nonexpansive maps

Gutiérrez, Armando W.

doi:10.1007/s11784-021-00896-8

Cited by 4 publications

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“…The isometric map T (x 1 , x 2 , ...) = (1, x 1 , x 2 , ...) of either ℓ 1 (N) or ℓ 2 (N), clearly has no fixed point in the usual sense. On the other hand, in the ℓ 1 (N) case it leaves the metric functional associated to (1, 1, 1, ...) invariant ( [GuK21])…”

Section: Introductionmentioning

confidence: 99%

“…In general, one conceivable path forward is to refine the theorem in this special case, and deduce that the metric functional must be a Busemann point, and that as such it has an associated closed linear subspace that must be invariant. A simpler discussion of the invariant subspace problem from a metric perspective without the space Pos can be found in [GuK21].…”

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A metric fixed point theorem and some of its applications

Karlsson

2022

Preprint

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A general fixed point theorem for isometries in terms of metric functionals is proved under the assumption of the existence of a conical bicombing. One special case provides a new mean ergodic theorem that in the Hilbert space case implies von Neumann's theorem. For CAT(0)-spaces and injective spaces the fixed point theorem is new for non-locally compact spaces, and implies the usual result for proper CAT(0)-spaces. For Banach spaces the therorem accomodates classically fixed-point-free isometric maps such as those of Kakutani, Edelstein, Alspach and Prus. Furthermore, from the main theorem together with some geometric arguments of independent interest, one can deduce that every bounded invertible operator of a Hilbert space admits a nontrivial invariant metric functional on the space of positive operators. This is a result in the direction of the invariant subspace problem although its full meaning is dependent on a future determination of such metric functionals.

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Section: Introductionmentioning

confidence: 99%

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confidence: 99%

A metric fixed point theorem and some of its applications

Karlsson

2022

Preprint

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Firm non-expansive mappings in weak metric spaces

Gutiérrez

Walsh

2022

Arch. Math.

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A Metric Fixed Point Theorem and Some of Its Applications

Karlsson

2024

Geom. Funct. Anal.

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A general fixed point theorem for isometries in terms of metric functionals is proved under the assumption of the existence of a conical bicombing. It is new for isometries of convex sets of Banach spaces as well as for non-locally compact CAT(0)-spaces and injective spaces. Examples of actions on non-proper CAT(0)-spaces come from the study of diffeomorphism groups, birational transformations, and compact Kähler manifolds. A special case of the fixed point theorem provides a novel mean ergodic theorem that in the Hilbert space case implies von Neumann’s theorem. The theorem accommodates classically fixed-point-free isometric maps such as those of Kakutani, Edelstein, Alspach and Prus. Moreover, from the main theorem together with some geometric arguments of independent interest, one can deduce that every bounded invertible operator of a Hilbert space admits a nontrivial invariant metric functional on the space of positive operators. This is a result in the direction of the invariant subspace problem although its full meaning is dependent on a future determination of such metric functionals.

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Comments on the cosmic convergence of nonexpansive maps

Cited by 4 publications

References 23 publications

A metric fixed point theorem and some of its applications

A metric fixed point theorem and some of its applications

Firm non-expansive mappings in weak metric spaces

A Metric Fixed Point Theorem and Some of Its Applications

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