I sense some sarcasm :) Luckily these things have nice geometrical
interpretations, let me try to clarify this better, hoping that I can
stimulate some curiosity on the subject.
Imagine a linear transformation in the 3-dimensional space; that
means that rotations, "flips", and scalings are allowed. For example
take a rotation: points that are in the axis of rotation remain on
that line. Likewise, think of the plane orthogonal to the axis of
rotation, then points that are on that plane rotate, but they stay on
that plane. The axis of rotation and its orthogonal plane are both
invariant subspaces.
In a 3-dimensional space, and in general in every odd-dimensional
space, every linear transformation has an invariant subspace. This is
not true for even-dimensional spaces, such as the 2d plane: a
45-degree rotation for example has no invariant subspace (except for {0}
and the whole space, of course). However, if you allow the coordinates
to be complex numbers, instead of real, then there always is an
invariant subspace, although it is harder to visualize.
To express this in linear algebra terms, every complex matrix has an
eigenvector. This is a fundamental fact that has applications pretty
much everywhere in mathematics.
The Invariant Subspace problem generalizes this statement to spaces
that are infinite-dimensional. Infinite-dimensional spaces are
infinitely messy, so we restrict our attention to spaces that have
some additional structure; for example Hilbert spaces, those addressed
in the paper, are those where a formula very similar to the
Pythagorean theorem is true.