A nice copy of a degenerate Lorentz-Marcinkiewicz space that implies the failure of the fixed point property

Abstract

Introducing the notion of asymptotically isometric copies inside Banach spaces, Dowling, Lennard and Turett made easier to detect failure of the fixed point property for nonexpansive mappings. Their tool was very usefull for indicating the failure. Since then, researchers have investigated alternative tools. Recently, Nezir introduced the notion of asymptotically isometric copies of $\ell^{1\boxplus 0}$. He noticed that a renorming of $\ell^1$ turns out to be a degenerate Lorentz-Marcinkiewicz space and using its structure he introduced his notion which implies the failure of the fixed point property for nonexpansive mappings. In this study, we introduce another notion which is derived from the structure of another degenerate Lorentz-Marcinkiewicz space and we show that detecting our new tool in Banach spaces will indicate the failure of the fixed point property for nonexpansive mappings.

Keywords

• fixed point property
• nonexpansive mappings
• Lorentz-Marcinkiewicz space
• degenerate Lorentz-Marcinkiewicz space
• asymptotically isometric copy of $\ell^1$

References

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