2010/11/01 by Alexandru Mihail, Mihail, Alexandru, Radu Miculescu +1
Mathematics · #28A80 #47B07 #54D05 #FOS: Mathematics #Functional Analysis (math.FA) #math.FA #msc:28A80 #msc:47B07 #msc:54D05
paper · pdf · doi:10.48550/arxiv.1011.0262
12 pages
arxiv created 2010/11/01 · arxiv updated 2010/11/02
In this paper we present a result which establishes a connection between the theory of compact operators and the theory of iterated function systems. For a Banach space X, S and T bounded linear operators from X to X such that ∥ S ∥, ∥ T ∥ <1 and w ∈ X, let us consider the IFS Sw=(X,f1,f2), where f1,f2:X → X are given by f1(x)=S(x) and f2(x)=T(x)+w, for all x ∈ X. On one hand we prove that if the operator S is compact, then there exists a family (Kn)n ∈ N of compact subsets of X such that A_Sw is not connected, for all w ∈ H- ∪ Kn. One the other hand we prove that if H is an infinite dimensional Hilbert space, then a bounded linear operator S:H → H having the property that ∥ S ∥ <1 is compact provided that for every bounded linear operator T:H→ H such that ∥ T ∥ <1 there exists a sequence (KT,n)n of compact subsets of H such that A_Sw is not connected for all w ∈ H- ∪ KT,n. Consequently, given an infinite dimensional Hilbert space H, there exists a complete characterization of the compactness of an operator S:H → H by means of the non-connectedness of the attractors of a family of IFSs related to the given operator.