Zoran Skoda
spectral theorem for bounded selfadjoint operators

Theorem (version from Kurepa’s functional analysis book)

For every bounded selfadjoint operator AA on Hilbert space HH there is a unique family (E λ,λR)(E_\lambda,\lambda\in\mathbf{R}) of orthogonal projectors on HH such that

(a) E λE μE_\lambda\leq E_\mu if λ<μ\lambda\lt\mu

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

(c) E λ=0E_\lambda = 0 if λ<m:=inf{(Ax,x)||x|=1,xH}\lambda \lt m := inf\{(Ax,x)\, | \,|x| = 1, x\in H\} and E μ=IE_\mu = I if λM:=sup{(Ax|x)||x|=1}\lambda\geq M:=sup\{(Ax|x)\, |\, |x|=1\}

(d) E λE_\lambda is a strong limit of polynomials of AA, i.e. for every λR\lambda\in\mathbf{R} there is a sequence (p nλ,nN)(p_{n\lambda},n\in\mathbf{N}) of polyomials such that

E λ=slimp nλ(A)E_\lambda = s-lim p_{n\lambda}(A)

(e) for every polynomial pp,

p(A)= m0 Mp(λ)dE λ p(A) = \int_{m-0}^M p(\lambda) dE_\lambda

and in particular

A= m0 MλdE λ 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 CC on HH commutes with AA iff commutes with E λE_\lambda for all λR\lambda\in\mathbf{R}

g) λ 0R\lambda_0\in\mathbf{R} is a regular point of operator AA iff λ(E λ)\lambda\mapsto (E_\lambda) is constant in some neighborhood of λ 0\lambda_0

h) the point spectrum σ p(A)\sigma_p(A) consists of points μσ(A)\mu\in\sigma(A) in which λE λ\lambda\mapsto E_\lambda is discontinuous, i.e. E μE μ0E_\mu \neq E_{\mu-0}

i) the continuous spectrum σ c(A)\sigma_c(A) of operator AA consists of those points μσ(A)\mu\in\sigma(A) where λE λ\lambda\to E_\lambda is continuous, i.e. E μ=E μ0E_\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.