Let $T$ be a (possibly unbounded) self-adjoint operator on a Hilbert space. Assume that we for some reason know that the point spectrum of $T$ consists of a finite number of eigenvalues $\lambda _1, \lambda _2, \ldots ,\lambda _N$ and that $T$ in addition to this has some continuous spectrum. Let $f(T)$ be defined for instance as in

http://en.wikipedia.org/wiki/Holomorphic_functional_calculus

(or in some other familiar way, e.g. the Cauchy-Green formula or using the Fourier transform) for appropriate functions $f$.

If $\Pi _p$ denotes the projection onto the point spectrum is it then true that $\Pi _p f(T)$ is a trace class operator? In that case, does it hold that $$ \operatorname{Tr} (\Pi _p f(T))= \sum _{j=1}^N f(\lambda _j) \quad ? $$