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In this paper we present a result which establishes a connection between thetheory of compact operators and the theory of iterated function systems. For aBanach space X, S and T bounded linear operators from X to X such that\parallel S \parallel, \parallel T \parallel <1 and w \in X, let us considerthe IFS S_{w}=(X,f_1,f_2), where f_1,f_2:X \rightarrow X are given byf_1(x)=S(x) and f_2(x)=T(x)+w, for all x \in X. On one hand we prove that ifthe operator S is compact, then there exists a family (K_{n})_{n \in N} ofcompact subsets of X such that A_{S_{w}} is not connected, for all w \in H-\cup K_{n}. One the other hand we prove that if H is an infinite dimensionalHilbert space, then a bounded linear operator S:H \rightarrow H having theproperty that \parallel S \parallel <1 is compact provided that for everybounded linear operator T:H\rightarrow H such that \parallel T \parallel <1there exists a sequence (K_{T,n})_{n} of compact subsets of H such thatA_{S_{w}} is not connected for all w \in H- \cup K_{T,n}. Consequently, givenan infinite dimensional Hilbert space H, there exists a completecharacterization of the compactness of an operator S:H \rightarrow H by meansof the non-connectedness of the attractors of a family of IFSs related to thegiven operator.

A characterization of compact operators via the non-connectedness of the attractors of a family of IFSs