spectral theorem for bounded selfadjoint operators

Theorem (version from Kurepa’s functional analysis book)

For every bounded selfadjoint operator $A$ on Hilbert space $H$ there is a unique family $(E_\lambda,\lambda\in\mathbf{R})$ of orthogonal projectors on $H$ such that

(a) $E_\lambda\leq E_\mu$ if $\lambda\lt\mu$

(b) function $\lambda\mapsto E_\lambda x$ is right continuous on $\mathbf{R}$ for each $x\in H$

(c) $E_\lambda = 0$ if $\lambda \lt m := inf\{(Ax,x)\, | \,|x| = 1, x\in H\}$ and $E_\mu = I$ if $\lambda\geq M:=sup\{(Ax|x)\, |\, |x|=1\}$

(d) $E_\lambda$ is a strong limit of polynomials of $A$, i.e. for every $\lambda\in\mathbf{R}$ there is a sequence $(p_{n\lambda},n\in\mathbf{N})$ of polyomials such that

$E_\lambda = s-lim p_{n\lambda}(A)$

(e) for every polynomial $p$,

$p(A) = \int_{m-0}^M p(\lambda) dE_\lambda$

and in particular

$A = \int_{m-0}^M \lambda dE_\lambda$

where the integral is the Riemann-Stieltjes integral taken in the sense of uniform convergence of operators.

Moreover, then there are additional properties

f) a bounded operator $C$ on $H$ commutes with $A$ iff commutes with $E_\lambda$ for all $\lambda\in\mathbf{R}$

g) $\lambda_0\in\mathbf{R}$ is a regular point of operator $A$ iff $\lambda\mapsto (E_\lambda)$ is constant in some neighborhood of $\lambda_0$

h) the point spectrum $\sigma_p(A)$ consists of points $\mu\in\sigma(A)$ in which $\lambda\mapsto E_\lambda$ is discontinuous, i.e. $E_\mu \neq E_{\mu-0}$

i) the continuous spectrum $\sigma_c(A)$ of operator $A$ consists of those points $\mu\in\sigma(A)$ where $\lambda\to E_\lambda$ is continuous, i.e. $E_\mu = E_{\mu-0}$

Last revised on March 16, 2011 at 16:13:58. See the history of this page for a list of all contributions to it.