This is an announcement for the paper "On the ``Multiple of the Inclusion Plus Compact'' problem" by George Androulakis and Frank Sanacory.

Abstract: The ``multiple of the inclusion plus compact problem'' which was posed by T.W.~Gowers in 1996 and Th.~Schlumprecht in 2003, asks whether for every infinite dimensional Banach space $X$ there exists a closed subspace $Y$ of $X$ and a bounded linear operator from $Y$ to $X$ which is not a compact perturbation of a multiple of the inclusion map from $Y$ to $X$. We give sufficient conditions on the spreading models of seminormalized basic sequences of a Banach space $X$ which guarantee that the ``multiple of the inclusion plus compact'' problem has an affirmative answer for $X$. Our results strengthen a previous result of the first named author, E.~Odell, Th.~Schlumprecht and N.~Tomczak-Jaegermann as well as a result of Th.~Schlumprecht. We give an example of a Hereditarily Indecomposable Banach space where our results apply. For the proof of our main result we use an extension of E.~Odell's Schreier unconditionality result for arrays.

Archive classification: Functional Analysis

Mathematics Subject Classification: 46A32, 47B07

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Submitted from: giorgis@math.sc.edu

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