On the "Multiple of the Inclusion Plus Compact" Problem

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.