Well, after some reading it turns out that the result is not trivial at all. Basically my proposition I wanted to show is called Gelfand-Maurin theorem, or Nuclear Spectral Theorem, and it states that for every number in the spectrum of A there is a generalised eigenvector, ie a distributional...
well,
\langle (A-zI)f_{n} \vert \phi \rangle \rightarrow 0
is definitely true, but it does not mean that
\langle f_{n}\vert \right.
converges weak-* to a nonzero functional, which would require
\forall \; \phi\in D(A)\quad \langle f_{n}\vert \phi \rangle=\langle T \vert \phi \rangle...
Hi All!
Preliminaries: Let H denote the Hilbert-space, and let A be a densely defined closed operator on it, with domain $D(A) \subset H$. On D(A) one defines a finer topology than that of H such way that f_n->f in the topology on D(A) iff both f_n->f and Af_n->Af in the H-topology. Let...