Let the operator $A$ be the generator of an analytic semigroup on a Banach space $X$. $Y$ is another Banach space embedded in $X$. $A_Y$, the part of $A$ in $Y$ is 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.

edityour original question, rather than leaving "corrections" as comments... – Matthew Daws Feb 25 '12 at 9:06