This paper is devoted to proving and discussing several consequences of the following
decomposition theorem:
Let
$A$
and
$B$
be closed denselydefined linear operators from the Banach space
$X$ to the Banach
space
$Y$ such
that
$D\left(B\right)\supseteq D\left(A\right)$,
$D\left({B}^{\ast}\right)\supseteq D\left({A}^{\ast}\right)$, the
range
$R\left(A\right)$ of
$A$ is closed, and the
dimension of the nullspace
$N\left(A\right)$
of
$A$ is finite.
Then
$X$ and
$Y$ can be decomposed
into direct sums
$X={X}_{0}\oplus {X}_{1}$,
$Y={Y}_{0}\oplus {Y}_{1}$, where
${X}_{1}$ and
${Y}_{1}$ are finite
dimensional,
${X}_{1}\subseteq D\left(A\right)$,
${X}_{0}\cap D\left(A\right)$ is dense
in
$X$, and
$\left({X}_{0},{Y}_{0}\right)$ and
$\left({X}_{1},{Y}_{1}\right)$ are invariant pairs of
subspaces for both
$A$
and
$B$. Let
${A}_{i}$ and
${B}_{i}$ be the
restrictions of
$A$
and
$B$ respectively
to
${X}_{i}$. For au
integers
$k$,
${\left({B}_{0}{A}_{0}^{1}\right)}^{k}\left(0\right)\subseteq R\left({A}_{0}\right)$,
and
$$dim{\left({B}_{0}{A}_{0}^{1}\right)}^{k}\left(0\right)=kdim\left({B}_{0}{A}_{0}^{1}\right)\left(0\right)=kdimN\left({A}_{0}\right).$$
Also, the action of
${A}_{1}$
and
${B}_{1}$
from
${X}_{1}$
to
${Y}_{1}$
can be given a certain canonical description.
