I'm trying to calculate the spectrum of the linear operator $T: L^2(0,1) \to L^2(0,1)$ given by $T(f) \to tf(t)$.
I've found a few facts about this operator but I'm still struggling to find the exact spectrum.
- The norm of $T$ is 1 and so we know the spectrum is contained in the unit disc.
- $T$ is self-adjoint and so the spectrum is real and also all the spectrum is approximate (the point spectrum is empty) - so I just need to look for approximate eigenvalues.
I know now that I should look for functions $f_n$ with unit norm such that $\int_0 ^1 |\lambda - t|^2 (f_n(t)^2) dt \to 0$ and from the above I just need to check $\lambda \in [-1,1]$.
I always find it difficult to find the approximate spectrum and I don't know how I can possibly go about finding such $f_n$ so I would really appreciate some tips on how to go about finding the approximate point spectrum more generally as well!