I am starting to read the basics of the theory of semigroups of operators, so this questions may be trivial, please forgive my ignorance on the matter.
On Pazy's book it's stated that
$A$ is a bounded operator $\iff$ $A$ generates a uniformly continuous-semigroup $(T(t))_{t\geq 0}$.
My question is, since an unbounded operator $A:X\to Y$ may be seen as a bounded operator $A:D(A)\to Y$, I am wondering, if I have for instance the Laplacian operator $\Delta$; this is an unbounded operator on $L^2(\mathbb R)$, but is it's bounded if we work on the Sobolev space $W^{2,2}$, then: The uniformly continuous-semigroup generated by $A$ is defined on $D(A)$? Otherwise I don't know how to interpret the statement on Pazy's book.
Thanks in advance.