Let $A$ be an unbounded selfadjouint operator in the Hilbert space $H$, having domain $D(A)$.
Denoting by $\sigma_A$ the spectrum of $A$, we have
$\inf \sigma_A \ = \ \inf_{u\in D(A),\|u\|=1} \ \langle u, A u\rangle$
My question is: is this still true if I take the infimum only over a dense subset (EDIT: here I mean a subspace, not just a subset) $C$ of $D(A)$ instead of the whole $D(A)$?
If $A$ was bounded this should be ok by continuity, but in the unbounded case?
EDIT: After some more reflection it seems to me that it is ok also in the unbounded case if $C$ is a core for $A$ (i.e. A is the closure of its restriction to $C$). If $C$ is not a core I start to doubt seriously that it is true in general (I'm thinking of Laplacian on bounded domain, the bottom of the spectrum is different in the Neumann and Dirichlet case...). Still I am not completely sure how to make this precise, and help is still appreciated.