Folner sequences and finite operators

This article analyzes Folner 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 Folner sequence (i.e. a Folner sequence of projections strongly converging to 1). In particular, any quasinormal, any subnormal, any hyponormal and any essentially normal operator has a proper Folner sequence. Moreover, we show that an operator is finite if and only if it has a proper Folner sequence or if it has a non-trivial finite dimensional reducing subspace. We also analyze the structure of operators which have no Folner sequence and give examples of them. For this analysis we introduce the notion of strongly non-Folner 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.

Removal of chromium (vi) in aqueous environments using cork and heat-treated cork samples from quercus cerris and quercus suber

Chromium (VI) removal and its reduction to chromium (III) from aqueous solution by untreated and heat-treated Quercus cerris and heat-treated Quercus suber black agglomerate cork granules was investigated. Initial screening studies revealed that among the sorbents tested, untreated Q. cerris and Q. suber black agglomerate are the most efficient in the removal of Cr(VI) ions and were selected for adsorption essays. Heat treatment adversely affected chromium adsorption and chromium (VI) reduction in Q. cerris cork. The highest metal uptake was found at pH 3.0 for Q. cerris and pH 2.0 for black agglomerate. The experimental data fitted the Langmuir model and the calculated qmax was 22.98 mg/g in black agglomerate and 21.69 mg/g in untreated Q. cerris cork. The FTIR results indicated that while in black agglomerate, lignin is the sole component responsible for Cr(VI) sorption, and in untreated Q. cerris cork, suberin and polysaccharides also play a significant role on the sorption. The SEM-EDX results imply that chromium has a homogenous distribution within both cork granules. Also, phloemic residues in Q. cerris granules showed higher chromium concentration. The results obtained in this study show that untreated Q. cerris and black agglomerate cork granules can be an effective and economical alternative to more costly materials for the treatment of liquid wastes containing chromium