Relationship between spectral projection and spectral measure

Question

Let $H$ be a Hilbert space and $T \in \mathcal{L}(H)$ be a normal operator. It is known, that there exists a spectral measure $E$ on $(\mathbb{C},\mathcal{B})$ (where $\mathcal{B}$ is the set of Borel measurable sets) such that $$ T = \int_{\mathbb{C}} z\,dE(z) $$ and that for every $\delta \in \mathcal{B}$ we have that $H(\delta):= E(\delta)H$ is a $T$-reducing subspace and that $\sigma(T\mid_{H(\delta)}) \subseteq \bar{\delta}$. However, for a decomposition $\sigma(T) = \delta \sqcup \delta'$ into two closed sets $\delta, \delta'$ we can also define $$ \mathcal{P}_{\delta} = \frac{1}{2\pi i}\int_{\Gamma} (\lambda I -T)^{-1}\, d\lambda $$ by the holomorphic functional calculus where $\Gamma$ encloses $\delta$ but not $\delta'$. One can easily show that $\mathcal{P}_{\delta}$ is a projection operator and since $\mathcal{P}_{\delta}$ commutes with $T$, the subspace $F(\delta) = \mathcal{P}_\delta H$ is a $T$-reducing subspace.\

Question: What is the relationship between these two concepts? What is the relationship between the subspaces $H(\delta)$ and $F(\delta)$? Is the spectral projection expressible through the spectral measure?

I would also appreciate any references where this is covered!

Answer 1

The answer to your question about $\delta,\delta'$ is that $$ \frac{1}{2\pi i}\oint_{\Gamma}(\lambda I-T)^{-1}d\lambda = E(\delta), $$ provided that $\Gamma$ is a positively oriented contour that encloses only the component $\delta$ of the spectrum. It's hard to isolate the components of the spectrum for a general normal operator. However, if $A$ is selfadjoint, then Stone's formula gives you a way to construct the spectral measure from a vector limit of the resolvent:

$$ \frac{1}{2}(E(a,b)+E[a,b])x\\=\lim_{\epsilon\downarrow 0}\frac{1}{2\pi i}\int_{a}^{b}((t-i\epsilon)I-A)^{-1}x-((t+i\epsilon I-A)^{-1}xdt $$ This was one of the earliest constructions of the spectral measure. Functional Analysis by Peter Lax details this approach to construct the spectral measure for (un)bounded self-adjoint linear operators on a Hilbert space.