Some properties of almost summing operators

DOI10.1016/J.JMAA.2021.125431arXiv2104.00435MaRDI QIDQ6364336

Joedson Santos, Renato Macedo

Publication date: 1 April 2021

Abstract: In this paper we extend the scope of three important results of the linear theory of absolutely summing operators. The first one was proved by Bu and Kranz in cite{BK} and it asserts that a continuous linear operator between Banach spaces takes almost unconditionally summable sequences into Cohen strongly $q$-summable sequences for any $qgeq2$, whenever its adjoint is $p$-summing for some $pgeq1$. The second of them states that $p$-summing operators with hilbertian domain are Cohen strongly $q$-summing operators ($1<p,q<infty$), this result is due to Bu cite{Bu}. The third one is due to Kwapie'{n} cite{Kwapien} and it characterizes spaces isomorphic to a Hilbert space using 2-summing operators. We will show that these results are maintained replacing the hypothesis of the operator to be $p$-summing by almost summing. We will also give an example of an almost summing operator that fails to be $p$-summing for every $1leqp<infty$.

Mathematics Subject Classification ID

Linear operators belonging to operator ideals (nuclear, (p)-summing, in the Schatten-von Neumann classes, etc.) (47B10)

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