Følner sequences and finite operators

This article analyzes Følner sequences of projections for bounded linear operators and their relationship to the class of finite operators introduced by Williams in the 70s. We prove that each essentially hyponormal operator has a proper Følner sequence (i.e., an increasing Følner sequence of projections strongly converging to 11). In particular, any quasinormal, any subnormal, any hyponormal and any essentially normal operator has a proper Følner sequence. Moreover, we show that an operator is finite if and only if it has a proper Følner sequence or if it has a non-trivial finite dimensional reducing subspace. We also analyze the structure of operators which have no Følner sequence and give examples of them. For this analysis we introduce the notion of strongly non-Følner 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.