Let $H$ be a Hilbert space and $A:D(A)\subset H\to H$ be a densely defined positive self adjoint unbounded operator. It can be prove that $\{e^{-tA}:t\geq 0\}$ is a $C_0$ or strongly continuous semigroup with $\|e^{-tA}\|\leq 1$, generated by $-A$, where $e^{-tA}x=\int_0^\infty e^{-t\nu }dE_\nu x$ and $\{E_\nu :\nu\geq0\}$ is the resolution of identity corresponding to the operator $A$. We know from semigroup theory that for $x\in D(A)$ (domain of $A$) $$\int_0^Te^{-(T-t)A} A x dt=(I-e^{-T A})x.$$ My question is that if we replace $x$ by a continuous function $g:[0,T]\to D(A)$, then $t\mapsto e^{-(T-t)A} Ag(t)$ is integrable or not.
It is clear that if $t \mapsto Ag(t)$ is integrable, then $t\mapsto e^{-(T-t)A} Ag(t)$ is integrable.