Eigenvalues of a linear operator

Jan 2010
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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

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Jun 2010
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What ideas have you had so far?
 
Jan 2010
39
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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

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Jun 2010
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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?
 
Jan 2010
39
0
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

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Jun 2010
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\(\displaystyle \displaystyle x\int_{0}^{1}(At+B)\,dt+\int_{0}^{1}t(At+B)\,dt=\lambda (Ax+B).\)

Turn the crank on the LHS...
 

HallsofIvy

MHF Helper
Apr 2005
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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.