The following question is motivated by the definition of spectral triples in noncommutative geometry. This question was split in the following parts:
- First: Could somebody give diverse examples of operators on Hilbert spaces, having compact resolvent?
Now, suppose one has certain algebra $A$ acting as operators on a Hilbert Space $X$. Certain self-adjoint operator $D$ on $X$ is imposed to satisfy, in particular, following axioms in order to be a deemed (generalized) Dirac operator, in an abstract sense:
- $[D,a]$ is bounded for each $a\in A$ and
$(D^2+1)^{-1/2}$ is a compact operator.
- Second: could somebody explain with an example, why conditions 1. and 2. tend to contradict each other?