  Finite rank perturbations of diagonalizable normal operators acting boundedly
on infinite dimensional, separable, complex Hilbert spaces are considered from
the standpoint of view of the existence of invariant subspaces. In particular,
if $T=D_\Lambda+u\otimes v$ is a rank-one perturbation of a diagonalizable
normal operator $D_\Lambda$ with respect to a basis $\mathcal{E}=\{e_n\}_{n\geq
1}$ and the vectors $u$ and $v$ have Fourier coefficients $\{\alpha_n\}_{n\geq
1}$ and $\{\beta_n\}_{n\geq 1}$ with respect to $\mathcal{E}$ respectively, it
is shown that $T$ has non trivial closed invariant subspaces provided that
either $u$ or $v$ have a Fourier coefficient which is zero or $u$ and $v$ have
non zero Fourier coefficients and
  $$ \sum_{n\geq 1} |\alpha_n|^2 \log \frac{1}{|\alpha_n|} + |\beta_n|^2 \log
\frac{1}{|\beta_n|} < \infty.$$
  As a consequence, if $(p,q)\in (0,2]\times (0,2]$ are such $\sum_{n\geq 1}
(|\alpha_n|^p + |\beta_n|^q )< \infty,$ it is shown the existence of non
trivial closed invariant subspaces of $T$ whenever
  $$(p,q)\in (0,2]\times (0,2]\setminus \{(2, r), (r, 2):\; r\in(1,2]\}.$$
  Moreover, such operators $T$ have non trivial closed hyperinvariant subspaces
whenever they are not a scalar multiple of the identity. Likewise, analogous
results hold for finite rank perturbations of $D_\Lambda$. This improves
considerably previous theorems of Foia\c{s}, Jung, Ko and Pearcy, Fang and Xia
and the authors on an open question explicitly posed by Pearcy in the
seventies.
