  The present work extends our short communication Phys. Rev. Lett. 95, 111102
(2005). For smooth marginally outer trapped surfaces (MOTS) in a smooth
spacetime we define stability with respect to variations along arbitrary
vectors v normal to the MOTS. After giving some introductory material about
linear non self-adjoint elliptic operators, we introduce the stability operator
L_v and we characterize stable MOTS in terms of sign conditions on the
principal eigenvalue of L_v. The main result shows that given a strictly stable
MOTS S contained in one leaf of a given reference foliation in a spacetime,
there is an open marginally outer trapped tube (MOTT), adapted to the reference
foliation, which contains S. We give conditions under which the MOTT can be
completed. Finally, we show that under standard energy conditions on the
spacetime, the MOTT must be either locally achronal, spacelike or null.
