We have the operator $T(f) = (pf')'$, where $p(x) = x^2 - 1$. The inner product is $\displaystyle (f,g) = \int_{-1}^1 f(x)g(x) dx$. How do we infer whether eigenfunctions corresponding to different eigenvalues are orthogonal?
2026-03-28 13:27:28.1774704448
How do we determine if an operator over real functions is normal?
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Integrating twice by parts, $$(T(f),g)=-(pf',g')=-(f',pg')=(f,T(g))$$ hence $T$ is self-adjoint.