Let the operator $A$ be the generator of an analytic semigroup on a Banach space $X$. 
Let $Y$ be another Banach space embedded in $X$. We consider$A\_Y$, the part of $A$ in $Y$, defined as the operator with domain


$$D(A\_Y) := \{ y \in D(A) \cap Y: Ay \in Y \}$$
and 
$$A\_Y \ y := Ay$$


Then it seems to me that $A\_Y$ is the generator of an analytic semigroup on $Y$. I didn't
find a proof, so I'm asking if someone can give a reference or counterexample if it is not true. 


(*Edit from Feb. 25 '12 comment*) The motivation for this question comes from Cauchy problems where the initial value is often taken to be in some intermediate space of the domain of the operator and the space X. This operator is usually analytic generator so I was wondering if this il also the case for its part.



The part of an operator as an analytic generator
This is wrong. Let us assume that $Y$ is a closed subspace of $X$ to clearify the problem. As Matthew Daws already said, you have to assume that the semigroup $(T(t))$ generated by $A$ leaves $Y$ invariant: suppose that $A\_Y$ indeed generates a (strongly continuous) semigroup $(S(t))$ on $Y$. Then for example the Yosida-approximation shows that $T\_Y(t) = S(t)$ for all $t$.


I have the impression that you have a concrete application in mind. So maybe you should reformulate the question in this concrete setting?


