  The nonmodal linear stability of the Schwarzschild black hole established in
Phys. Rev. Lett. 112 (2014) 191101 is generalized to the case of a nonnegative
cosmological constant $\Lambda$. Two gauge invariant combinations $G_{\pm}$ of
perturbed scalars made out of the Weyl tensor and its first covariant
derivative are found such that the map $[h_{\alpha \beta}] \to \left( G_-
\left([h_{\alpha \beta}] \right), G_+ \left([h_{\alpha \beta}] \right) \right)$
with domain the set of equivalent classes $[h_{\alpha \beta}]$ under gauge
transformations of solutions of the linearized Einstein's equation, is
invertible. The way to reconstruct a representative of $[h_{\alpha \beta}]$ in
terms of $(G_-,G_+)$ is given. It is proved that, for an arbitrary perturbation
consistent with the background asymptote, $G_+$ and $G_-$ are bounded in the
the outer static region. At large times, the perturbation decays leaving a
linearized Kerr black hole around the Schwarzschild or Schwarschild de Sitter
background solution. For negative cosmological constant it is shown that there
is a choice of boundary conditions at the time-like boundary under which the
Schwarzschild anti de Sitter black hole is unstable. The root of
Chandrasekhar's duality relating odd and even modes is exhibited, and some
technicalities related to this duality and omitted in the original proof of the
$\Lambda=0$ case are explained in detail.
