Existence of solution of some functional equation involving integral

Question

I want to prove that there exists always solutions to this equation $$g(x) = \int\limits_a^b {f(s,x)ds} $$ where $g \in {L^2}(0,1)$ is given and $f \in {L^2}((a,b) \times (0,1))$ is the uknown, $0<a<b<1$ are just to positive constants. If I consider the operator $$\eqalign{ & T:{L^2}((a,b) \times (0,1)) \to {L^2}(0,1) \cr & f \to \int\limits_a^b {f(s,x)ds} \cr} $$ the surjectivity peoblem turn to the existence of a positive constant $c$ such that $${\left\| {{T^*}h} \right\|_{{L^2}(0,1)}} \geqslant c{\left\| h \right\|_{{L^2}((a,b) \times (0,1))}}$$ The formal ajoint of $T$ is given by ${T^*}h =h$, therefore, we can write $${\left\| {{T^*}h} \right\|_{{L^2}(0,1)}} = \int\limits_0^1 {{h^2}(s)ds = } \int\limits_a^b {\int\limits_0^1 {{h^2}(s)dsdx = } } \frac{1}{{b - a}}{\left\| h \right\|_{{L^2}((a,b) \times (0,1))}}$$. Am I write? thank you.