# Eigenvalues of a linear operator

#### featherbox

T is a bounded linear operator mapping from C[0,1] to C[0,1] with the two norm.
Find the non-zero eigenvalues of T, where

$$\displaystyle (Tf)(x)=\int_{a}^{b} f(t)(x+t)dt$$

You may assume all eigenvectors for non-zero eigenvalues are of the form $$\displaystyle f(x)=Ax+B$$

#### Ackbeet

MHF Hall of Honor
What ideas have you had so far?

#### featherbox

What ideas have you had so far?
So far I have $$\displaystyle (Tf)(x)=\lambda x$$ then

$$\displaystyle x\int_{0}^{1}f(t)dt+\int_{0}^{1}tf(t)dt=\lambda (Ax+B)$$

but then I don't know how to get the $$\displaystyle \lambda$$

#### Ackbeet

MHF Hall of Honor
This is a somewhat puzzling problem, for reasons that will become clear later. Why not plug in for f(t) under your integrals like you did on the RHS?

#### featherbox

This is a somewhat puzzling problem, for reasons that will become clear later. Why not plug in for f(t) under your integrals like you did on the RHS?
It is plugged in, isn't it? I don't understand what you mean. Could you elaborate please?

#### Ackbeet

MHF Hall of Honor
$$\displaystyle \displaystyle x\int_{0}^{1}(At+B)\,dt+\int_{0}^{1}t(At+B)\,dt=\lambda (Ax+B).$$

Turn the crank on the LHS...

#### HallsofIvy

MHF Helper
To say that f is an eigenvector of linear transformation T means that it is $$\displaystyle Tf= \lambda f$$. Ackbeet is saying that since you are told that you can write the eigenvector as "Ax+ B" you should use that on both sides of the equation.