Space of vector valued Lipschitz functions and the Daugavet property

Abraham Rueda Zoca (U of Granada)

Mar 19. 2025, 09:00 — 09:45

Recall that a Banach space $X$ has the \textit{Daugavet property} if, given any rank-one linear and continuous operator $T:X\longrightarrow X$ it follows that $$\Vert T+\mathrm{I}\Vert=1+\Vert T\Vert,$$ where $\mathrm{I}:X\longrightarrow X$ stands for the identity operator.   In the context of the Lipschitz free space over a complete metric space, it is known that $\mathcal F(M)$ has the Daugavet property if, and only if, the metric space $M$ is \textit{length}, i.e., if for every pair of distinct points $x,y\in M$ then $d(x,y)$ equals the inf. of the length of all the rectifiable curves joininig $x$ and $y$ [1] and if, and only if, its dual $\Lip(M)$ has the Daugavet property.   In this talk we will analyse the question when $\Lip(M,X)$ has the Daugavet property for every Banach space $X$. The results of this talk are part of the preprint \cite{mr} in collaboration with R. Medina [2].

 

[1] L. García-Lirola, A. Procházka and A. Rueda Zoca, A characterisation of the Daugavet property in spaces of Lipschitz functions, J. Math. Anal. Appl. 464 (2018), 473–492.

[2] R. Medina and A. Rueda Zoca, A characterisation of the Daugavet property in spaces of vector-valued Lipschitz functions, preprint. Available at ArXiV.org with reference arXiv:2305.05956.

Further Information
Venue:
ESI Boltzmann Lecture Hall
Recordings:
Recording
Associated Event:
Structures in Banach Spaces (Workshop)
Organizer(s):
Antonio Aviles (U Murcia)
Vera Fischer (U of Vienna)
Grzegorz Plebanek (U of Wroclaw)
Damian Sobota (U of Vienna)