Shorted operators: An application in potential theory

✍ Scribed by Volker Metz

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 692 KB
• Volume: 264
• Category: Article
• ISSN: 0024-3795
• DOI: 10.1016/s0024-3795(96)00303-5

✦ Synopsis

On nested fractals a "Laplacian" can be constructed as a scaled limit of difference operators. The appropriate scaling and starting configuration are given by a nonlinear, finite dimensional eigenvalue problem. We study it as a fixed point problem using Hilbert's projective metric on cones, a nonlinear generalization of the Perron-Frobenius theory of nonnegative matrices. The nonlinearity arises from a map known as the shorted operator. Potential theoretic notions and results apply to it, since it acts on a cone of discrete "Laplacians'" or difference operators. Usually, ~P is considered on the larger cone of positive semidefinite operators. We are able to take advantage of the more specific structure of the reduced domain because several properties of dp are local. Results are possible with respect to continuity, concavity, the Fr~chet derivative, invariant subcones, the geometry of these cones, and the contraction of Hilbert's metric.