I want to find the kernel and range of:
$$T:C([0,1])\rightarrow C([0,1])$$
$$Tf(x)=\int_a^b(x-t)f(t)dt$$
I know I'm looking for $f(x)$ that satisfy:
$$x\int_a^bf(t)dt-\int_a^btf(t)dt=0$$ but I'm looking for more information about $f(x)$. I thought about a change of variables to shift my integrals to $\int_0^1$ and then using a odd or even extension and looking at fourier modes, but I haven't had any luck.
The null space of $T$ consists of all functions $f \in C[0,1]$ that are orthogonal to $1,t$, meaning that $$ \int_a^b f(t)dt = \int_a^b t f(t)dt = 0 $$
The range of $T$ is the two-dimensional subspace $[\{1,t\}]$.
For every $g\in C[0,1]$, you can construct an element $f$ in the null space of $T$ by subtracting away an orthogonal projection onto $[\{1,t\}]$: $$ f= g(x)-\frac{\int_a^b g(x)dx}{b-a}-\frac{\int_a^b g(x)(x-\frac{b+a}{2})dx}{\int_a^b (x-\frac{b+a}{2})^2dx}(x-\frac{b+a}{2}). $$