TITLE: Is this closed subspace of Fréchet space complemented
QUESTION [6 upvotes]: In the hope of completing the rich tapestry of complemented (or not) topological vector subspaces, I would like to know (maybe it is immediate for specialists) 
whether the space of analytic functions is complemented  within the space of infinitely differentiable ones. I begin with the one-variable case 
...  and make this precise. 

Let $\Omega\subset \mathbb{C}$ be an open subset. We consider
  $$
H(\Omega)=C^\omega(\Omega;\mathbb{C})\subset C^\infty(\Omega;\mathbb{C})
$$
  the large one being endowed with the standard topology defined by the seminorms
  $$
p_{\,n,B}=sup_{\ 0\leq |\alpha|\leq n\atop t\in B}|D^\alpha(f)[t]|\ . 
$$
  where $n\in \mathbb{N}, \alpha\in \mathbb{N}^2$, $B$ is a relatively compact open subset of $\Omega$ and the bi-indexed derivative is 
  $$
D^\alpha:=(\frac{\partial}{\partial x})^{\alpha[1]}(\frac{\partial}{\partial y})^{\alpha[2]}\ .
$$ 
  I know that the subspace $H(\Omega)=C^\omega(\Omega;\mathbb{C})$ is complete and then closed for this (standard) topology. My question is the following
Q) Is there a known closed complement of it i.e. a decomposition
  $$
C^\infty(\Omega;\mathbb{C})=C^\omega(\Omega;\mathbb{C})\oplus W=H(\Omega)\oplus W
$$
  where $W$ is closed ? (maybe the projector is an integro-differential operator ?) at least for some particular domains $\Omega$ ?

Remark i) This question is a reformulation of 
this one
in MSE where it did not receive a complete answer. 
ii) With the given topology, $C^\infty(\Omega;\mathbb{C})$ and $H(\Omega)=C^\omega(\Omega;\mathbb{C})$ are m-convex Fréchet algebras. Maybe (if possible) $W$ could have some algebraic structure (ideal ?).

REPLY [6 votes]: $H(\Omega) $ is not complemented in $C^\infty (\Omega)$ e.g. for the unit disc in $\mathbb C $. This follows from the structure theory of Frechet spaces: The space of smooth functions is isomorphic to $s^\mathbb N$ and has a certain property  (DN$_{loc}$) of Vogt. If $H (\Omega) $ were complemented it would also have this property and hence   even property (DN) because it has continuous norms.  But this is not true for power series spaces of finite type.   
I don't have any literature at hand.  The book of Meise and Vogt is a good starting point. The local condition is in a more recent article of Vogt.