Non-point spectrum for diagonalisable self-adjoint unbounded operator

Given a (separable) Hilbert space H and an unbounded densely defined linear operator $$T:{\cal D}(T) \to$$H such that $${\cal D}$$ is diagonalizable (it means $$\exists$$ an O.N.B. of H such that all basis elements are eigenvectors of $$T$$). Is it possible for $$T$$ to have non-point spectrum, I mean, can exist a $$\lambda \in \sigma(T)$$ such that $$\lambda$$ is not an eigenvalue of $$T$$?

Take $$T$$ to be the inverse of a bounded/continuous, self-adjoint operator with eigenvalues (an orthonormal basis) all rationals between $$0$$ and $$1$$. Then $$T$$ has an orthonormal basis of eigenvectors, with eigenvalues all rationals above $$1$$. Spectra are closed...