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F\olner sequences and finite operators

This article analyzes F\olner sequences of projections for bounded linear operators and their relationship to the class of finite operators introduced by Williams in the 70ies. We prove that each essentially hyponormal operator has a proper F\olner sequence (i.e. a F\olner sequence of projections strongly converging to 1). In particular, any quasinormal, any subnormal, any hyponormal and any essentially normal operator has a proper F\olner sequence. Moreover, we show that an operator is finite if and only if it has a proper F\olner sequence or if it has a non-trivial finite dimensional reducing subspace. We also analyze the structure of operators which have no F\olner sequence and give examples of them. For this analysis we introduce the notion of strongly non-F\olner operators, which are far from finite block reducible operators, in some uniform sense, and show that this class coincides with the class of non-finite operators.