The Spectral Theorem

Abstract

This chapter offers a proof of the spectral theorem for self-adjoint operators A by Hilbert space intrinsic methods. The starting point is the so-called geometric characterization of self-adjointness in terms of the subspaces of controlled growth $F(A,r)\subset D^{\infty}(A)=\cap_n D(A^n)$ of the operator, i.e., for a closed symmetric operator A and $0\leq r <s$ one has $r\left\Vert{x}\right\Vert\leq \left\Vert{Ax}\right\Vert\leq s\left\Vert{x}\right\Vert$ for all $x \in F(A,s)\cap F(A,r)^{\perp}$ and such an operator is self-adjoint if, and only if, $\cup_n F(A,n)$ is dense in the Hilbert space $\mathcal{H}$. Then spectral families $E_t, t \in \mathbb{R}$ are introduced as monotone, right-continuous, normalized projection-valued functions on $\mathbb{R}$.They are characterized by corresponding properties of the family of their ranges $H_t,t\in \mathbb{R}$. Next we define integrals of continuous functions with respect to a spectral family and study their basic properties. For a self-adjoint operat or $A\geq 0$ the spectral family E _t is defined by $\rm{ran}\, E_t =F(A,t)$ for $t \geq 0$ and $=\left\{0\right\}$ for $t<0$. Through various approximations the spectral representation $A=\int t \mathrm{d} E_t$ follows. The general case can be reduced to the case of positive A. As applications of this approach the maximal self-adjoint part of a closed symmetric operator is easily determined. Furthermore convenient sufficient conditions can be given under which a closed symmetric operator is essentially self-adjoint.