Find eigenvalues for $T(f) = \int_{-\infty}^x tf(t)dt$ Let $V$ be the linear space of all functions continuous on $(-\infty, \infty)$ and such that that the integral $\int_{-\infty}^x tf(t)\,dt$ exists. If $f \in V$, let $g=T(f)$ be defined as $g(x) = \int_{-\infty}^x tf(t)\,dt$. Prove that every $\lambda < 0$ is an eigenvector and determine the eigenfunctions corresponding to $\lambda$.
We know that $T(f) = \int_{-\infty}^x tf(t)\,dt = \lambda f(x)$. (Not sure if this is right so far.) So know what do I do? 

as cameron suggests, take the equation $$ \int_{-\infty}^x tf(t)\, dt = \lambda f(x) \tag 1$$
first, we will deal with the case $\lambda = 0.$
diffrentiang $(1)$ gives $xf(x) = 0$ implying $f \equiv 0$ contradicting that $f$ is an eigenfunction.
now we are assuming $\lambda \neq 0.$
differentiating $(1),$ we get $$ xf(x) = \lambda f'(x).$$ separate the variables to get $$\frac{df}{f} = \frac 1\lambda x\, dx $$ gives $$f = Ce^{x^2/\lambda}  $$ now the boundary condition $\lim_{x \to -\infty}f(x) = 0$ implies $\lambda < 0.$
