We prove that if$T$is a strictly singular one-to-one operator defined on an infinite dimensional Banach space$X$, then for every infinite dimensional subspace$Y$of$X$there exists an infinite dimensional subspace$Z$of$X$such that$Z∩Y$is infinite dimensional,$Z$contains orbits of$T$of every finite length and the restriction of$T$to$Z$is a compact operator.

We construct a 2-dimensional complex manifold$X$which is the increasing union of proper subdomains that are biholomorphic to ℂ^{2}, but$X$is not Stein.

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