# How to get spectrum and eigenvectors of an integral operator?

This question is a series of How to construct the Singular Value Decomposition(SVD) of an operator P?

Briefly, there's an operator $P: L^2([-1,1]) \rightarrow L^2([3,5])$: $$f(x) = P(\phi)(x) = \int_{-1}^{1} \log(x-t)\phi(t)dt$$ and its adjoint $P^*$: $$P^*(f)(t) = \int_3^5\log(x-t)f(x)dx$$

Let $T = P^*P$, then T is an compact selfadjoint operator on $L^2([-1,1])$: $$T(\phi)(t) = \int_3^5\log(x-t) \int_{-1}^1\log(x-\tau)\phi(\tau)d\tau dx$$

How could I get the eigenvalues and corresponding eigenvectors of $T$?

@Dirk says *there is an orthonormal basis of eigenvectors of $P^*P$ (with nonnegative eigenvalues $\sigma_k$)*. I read functional analysis textbook but failed to find details on how to get them.

PS: My ultimate goal is to find the SVD of this kind of operator.

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FWIW: saying "SVD decomposition" is a bit like saying "I'll input my PIN number into the ATM machine"... –  Ｊ. Ｍ. Sep 18 '11 at 15:24
Thank you. I've corrected the post :) –  Yao Jin Sep 18 '11 at 15:29