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Strongly bounded operators on Crc(X,E)Crc(X,E) with the strict topology βσβσ

Research paper by Marian Nowak

Indexed on: 23 Jun '16Published on: 22 Jun '16Published in: Indagationes Mathematicae



Abstract

Let <img height="9" border="0" style="vertical-align:bottom" width="13" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0019357716300271-si33.gif">X be a completely regular Hausdorff space, and <img height="9" border="0" style="vertical-align:bottom" width="12" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0019357716300271-si21.gif">E and <img height="9" border="0" style="vertical-align:bottom" width="11" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0019357716300271-si22.gif">F be Banach spaces. Let <img height="13" border="0" style="vertical-align:bottom" width="59" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0019357716300271-si1.gif">Crc(X,E) be a space of all continuous functions <img height="12" border="0" style="vertical-align:bottom" width="67" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0019357716300271-si7.gif">f:X→E such that <img height="13" border="0" style="vertical-align:bottom" width="32" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0019357716300271-si36.gif">f(X) is a relatively compact set in <img height="9" border="0" style="vertical-align:bottom" width="12" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0019357716300271-si21.gif">E, equipped with the strict topology <img height="13" border="0" style="vertical-align:bottom" width="16" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0019357716300271-si10.gif">βσ. We study <img height="13" border="0" style="vertical-align:bottom" width="66" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0019357716300271-si11.gif">(βσ,‖⋅‖F)-continuous strongly bounded operators <img height="13" border="0" style="vertical-align:bottom" width="115" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0019357716300271-si100.gif">T:Crc(X,E)→F. In particular, we establish the relationships between <img height="13" border="0" style="vertical-align:bottom" width="66" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0019357716300271-si11.gif">(βσ,‖⋅‖F)-continuous strongly bounded operators and weakly compact (resp. weakly precompact; unconditionally converging; completely continuous; weakly completely continuous) operators <img height="13" border="0" style="vertical-align:bottom" width="115" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0019357716300271-si100.gif">T:Crc(X,E)→F. In particular, it is shown that if <img height="9" border="0" style="vertical-align:bottom" width="12" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0019357716300271-si21.gif">E is a Schur space, then the space <img height="13" border="0" style="vertical-align:bottom" width="91" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0019357716300271-si16.gif">(Crc(X,E),βσ) has the strict Dunford-Pettis property.