I was studying abstract inverse source problem of an abstract heat equation in approach of semi group theory. There I am unable to find the reason of existence of inverse of an operator that i have described below.
Let $X$ be a Banach space $\lbrace {T(t)}\rbrace _{t\geq 0}$ be a $C_0$ semi group of contraction of bounded linear operator on $X$ generated by a dissipative operator $A:D(A)\rightarrow X $ , where $D(A)\subset X$ and dense in $X$ . Let $t_0$ is a fixed positive real number. Then $(T(t_0)-I)^{-1}$ exists.
How do i prove the existence? I tried as follows.
As ${T(t)}$ is a $C_0$ semi group of contraction, $||T(t)||\leq \exp(wt)$ for some $w\geq {0}$. Then $||T(t_0)x-x||\leq |\exp(wt_0)-1|*||x||$ for all $x$ in $ X$. Now if I can able to prove that range of $(T(t_0)-I) $ is $X$, then the operator $(T(t_0)-I)$ is invertible.