Let $T(t)$ be a $C_0$-semigroup on Banach space $X$, and $A$ its generator. By Lumer-Philipps theorem we know that if $A$ is densely defined and m-dessipative operator then it generates a $C_0$-semigroup of contractions, I.e., $$\|T(t)\| \leq 1, \quad \forall t \geq 0.$$ My question here: is there any theorem which gives conditions to generate a strictly contraction $C_0$-semigroup? That is, $$\|T(t)\| < 1, \quad \forall t > 0.$$ For example, for $A=\Delta$ the Dirichlet Laplacian, is the associated semigroup strictly contractive? Can we calculate the norm $\|T(t)\|$ in this case? Or at least, Is $I-T(t)$ invertible? If this is true for the Dirichlet laplacian, can we obtain the same result for second order elliptic operator?
Your conditions is for contraction semigroups equivalent to have uniform exponential stability, i.e., to have growth bound less than zero, see Proposition V.1.7. in
Engel, Klaus-Jochen; Nagel, Rainer, One-parameter semigroups for linear evolution equations, Graduate Texts in Mathematics. 194. Berlin: Springer. xxi, 586 p. (2000). ZBL0952.47036.
Assuming you work in a Hilbert space, you can reformulate the Lumer.Phillips theorem for your case as $$\Re \langle Ax,x\rangle \leq \omega<0$$ to get a sufficient condition. For many examples you can test this by applying some version of Green's theorem.
For second order elliptic operators on bounded domains it boils down to look at the eigenvalues. If they are negative, you have exponential stability.
ADDED after comments: For uniformly elliptic operator it all depends on the boundary conditions. Neumann boundary conditions imply that you do not have all negative eigenvalues.
If you are interested in the spectral theory of differential operators, there are many good books. I liked the one
Davies, E. B., Spectral theory and differential operators, Cambridge Studies in Advanced Mathematics. 42. Cambridge: Cambridge Univ. Press. ix, 182 p. (1995). ZBL0893.47004.
very much, it is a good start.