  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.
