On quasi-compactness of operator nets on Banach spaces

Volume 203 / 2011

Eduard Yu. Emel'yanov Studia Mathematica 203 (2011), 163-170 MSC: Primary 47A35; Secondary 47B99, 47L05, 47S99. DOI: 10.4064/sm203-2-3


The paper introduces a notion of quasi-compact operator net on a Banach space. It is proved that quasi-compactness of a uniform Lotz–Räbiger net $(T_\lambda )_{\lambda }$ is equivalent to quasi-compactness of some operator $T_\lambda $. We prove that strong convergence of a quasi-compact uniform Lotz–Räbiger net implies uniform convergence to a finite-rank projection. Precompactness of operator nets is also investigated.


  • Eduard Yu. Emel'yanovDepartment of Mathematics
    Middle East Technical University
    06531 Ankara, Turkey
    Sobolev Institute of Mathematics
    630090 Novosibirsk, Russia

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