Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



The ideal of $ p$-compact operators and its maximal hull

Author: Albrecht Pietsch
Journal: Proc. Amer. Math. Soc. 142 (2014), 519-530
MSC (2010): Primary 47B07, 47B10; Secondary 46B22, 47L20
Published electronically: October 17, 2013
MathSciNet review: 3133993
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Abstract: Many recent papers have been devoted to the ideal of $ p$-compact operators, denoted by $ \mathfrak{K}_p$. In this note, we will revisit their main results by very simple proofs that are based on the general theory of operator ideals as developed in the author's monograph more than 30 years ago. In several cases the outcome is even better. Moreover, the ideal of all operators with $ p$-summing duals is identified as the maximal hull of $ \mathfrak{K}_p$. As a byproduct, we show that $ \mathfrak{K}_p^{\rm max}$ is stable under the formation of projective tensor products.

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Albrecht Pietsch
Affiliation: Friedrich-Schiller-Universität Jena, Fakultät für Mathematik und Informatik, Ernst-Abbe-Platz 2, D-07743 Jena, Germany
Address at time of publication: Biberweg 7, D-07749 Jena, Germany

Keywords: $p$-compact operators, quasi $p$-nuclear operators, absolutely $p$-summing operators, $(s,p)$-mixing operators, maximal operator ideals, surjective operator ideals
Received by editor(s): July 11, 2011
Received by editor(s) in revised form: March 4, 2012, and March 18, 2012
Published electronically: October 17, 2013
Communicated by: Marius Junge
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.