Generalised eigenvalues in the residual spectrum

Question

For an operator $T$ on a Hilbert space the residual spectrum is defined as the set of complex numbers $\lambda$ for which $(T-\lambda I)^{-1}$ exists but is not densely defined. The question is: could there be a complex number in the residual spectrum which is a generalised eigenvalue? By generalised eigenvalue I mean that there exists a sequence $\{x_n\}$ of unit vectors that satisfy: $$ (T-\lambda I)x_n \longrightarrow 0. $$ I was thinking about this from a physics point of view (discarding the fact that observables are assumed to be self adjoint) and wondering if one wanted to make an "observable" out of something which is hermitian but doesn't have an empty residual spectrum (like momentum on $L^2(0,\infty))$ would the problem be that for such operators the system could in principle collapse to some state with a complex spectral value associated to it? This spectral value would have to be in the residual spectrum because the continuous and point spectra of hermitian operators must be real.

Answer 1

Let $S : \mathcal H\to\mathcal H$ be any linear operator with non-closed range and $\ker S = \{0\}$ (for example $Sf = t\cdot f$ on $L^2(0,1)$ satisfies this). Now, let $M$ be any closed, proper infinite-dimensional subspace of $\mathcal H$ and let $V : \overline{\operatorname{ran}S}\to M$ be unitary. Then $T := VS$ satisfies the requirements.