The spectrum of uniformly Lipschitz mappings. (English) Zbl 0661.47048

We consider mappings \(\phi\) of a normed space E into itself which have the properties that \(\phi (0)=0\) and that there is a constant \(K>0\) with \(\| \phi (x)-\phi (y)\| \leq K\{x-y\|\) for all x, y in E. These mappings we call uniformly Lipschitz. Bounded linear mappings are examples of such maps, but they form a very small subspace of the whole. The collection of all uniformly Lipschitz mappings is denoted by \(\ell (E)\), which may be normed in a natural way. This space is complete if E is complete. Although \(\ell (E)\) is closed under composition it fails to be a Banach algebra.
We give a natural definition of the spectrum \(\sigma\) (\(\phi)\) of \(\phi\in \ell (E)\) and establish that if E is complete then \(\sigma\) (\(\phi)\) is bounded and closed, and that it is non-empty if E is one- dimensional. The non-emptiness in general is an open question.
The spectrum is calculated for certain uniformly Lipschitz maps on the complex numbers and on Banach lattices, and applied to certain mappings related to strong summability theory.
Reviewer: I.J.Maddox


46H05 General theory of topological algebras
46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces