I can't seem to find anywhere a discussion on whether or not the idea of a zero eigenvalue (value in the point spectrum) implies the inverse doesn't exists for an infinite dimensional operator.
We know for a finite operator, this is the case (a 0 eigenvalue means a matrix is not invertible). I am wondering if this same condition (or some other condition on the eigenvalues or spectrum) gives us that the inverse of an infinite dimensional operator doesn't exist.
Best Answer
Yes. Having an eigenvalue of $0$ is equivalent to $\operatorname{ker} (T - 0I)$ is non-trivial, which is equivalent to $T$ not being injective. What breaks down is that $0$ not being an eigenvalue only implies $T$ is injective, not that $T^{-1}$ exists, as injectivity and bijectivity are no longer equivalent for linear operators on infinite-dimensional spaces.