Spectrum of an integral operator

Question

I am trying to find the spectrum of the operator $Tf(x)=\int_{0}^{1}(1+x^2y^2)f(y)dy$ in $L^2(0,1)$.

Thanks a lot for any help.

Answer 1

Hint: $Tf=\lambda f $ is equivalent to $$\lambda f(x)=\int_0^1f(y)dy+x^2\int_0^1y^2f(y)dy$$ Hence either

• $\lambda\ne0$ and $f(x)=Ax^2+B$; then the equation above becomes $$\lambda(Ax^2+B)=\tfrac{1}{3}A+B+(\tfrac{1}{5}A+\tfrac{1}{3}B)x^2$$ Comparing coefficients gives $\begin{pmatrix}\frac{1}{5}&\frac{1}{3}\\\frac{1}{3}&1\end{pmatrix}\begin{pmatrix}A\\B\end{pmatrix}=\lambda\begin{pmatrix}A\\B\end{pmatrix}$, so $\lambda$ equals the eigenvalues $\lambda,\mu$ of this matrix and $(A,B)$ their eigenvectors.
• $\lambda =0$ then $\int_0^1f=0=\int_0^1y^2f(y)dy$. In this case, there are an infinite number of linearly independent solutions for $f$, in fact any function of the type $f(x):=g(x)-(A+Bx^2)$ for $g\in L^2[0,1]$ can satisfy these two equations if the right $A,B$ are chosen.

Hence the spectrum is $\{0,\lambda,\mu\}$.