I encountered upon reading a functional analysis book that for unbounded self-adjoint operators on a Hilbert space, commutativity is defined in terms of the corresponding bounded operators, for example, the spectral projections.
What exactly is the spectral projection? Could anyone explain briefly? I know bits of functional calculus.
Because you know something about the holomorphic functional calculus, then I can tell you how to construct the spectral measure for an unbounded self-adjoint operator $A: \mathcal{D}(A)\subset\mathcal{H}\rightarrow\mathcal{H}$ from the resolvent operator $(\lambda I-A)^{-1}$, which is defined and is bounded everywhere for $\lambda\in\mathbb{C}\setminus\mathbb{R}$. The spectral measure $E[a,b]$ of intervals $[a,b]$, $[a,b)$, $(a,b]$ or $(a,b)$ can be directly constructed through the resolvent as a type of generalized residue: $$ \frac{1}{2}(E[a,b]+E(a,b))f= \lim_{\epsilon\downarrow 0}\frac{1}{2\pi i}\int_a^b ((u-i\epsilon)I-A)^{-1}f-((u+i\epsilon)I-A)^{-1}f du. $$ If you define $E(\lambda)f=\lim_{a\downarrow -\infty}E[a,\lambda]$, then you can write $$ F(A)f=\int_{-\infty}^{\infty}F(\lambda)dE(\lambda)f. $$ And there is a type of Parseval identity as well: $$ \|F(A)f\|^2 = \int_{-\infty}^{\infty}|F(\lambda)|^2d\|E(\lambda)f\|^2 $$