I'm looking at the convergence (when blowing up the metric) of the spectrum of a self-adjoint operator $P$ that acts on differential forms of a 3-dimensional closed manifold M. Let $\lambda$ be a complex (not real) number. I need to prove the following inequality $$\|(\lambda-P)\alpha\|_0^2\geq |Im(\lambda)|^2\|\alpha\|^2_0+\|P\alpha\|_0^2$$ with respect to the $L^2$-norm. In some sense, the real part of $\lambda$ eats the double inner product, i.e. $$|Re(\lambda)|^2\|\alpha\|^2_0\geq 2(P_\epsilon\alpha,\lambda\alpha)_0.$$ Have you seen this happening somewhere else? Is there a general property of the spectrum of self-adjoint operators?
Asked
Active
Viewed 63 times
