An Invitation to Random Schr¨odinger operators - FernUniversität in ...

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(A − z 1 ) −1 − (A − z 2 ) −1 = (z 1 − z 2 ) (A − z 1 ) −1 (A − z 2 ) −1 (3.17)
and, if D(A) = D(B),
= (z 1 − z 2 ) (A − z 2 ) −1 (A − z 1 ) −1 (3.18)
(A − z) −1 − (B − z) −1 = (A − z) −1 (B − A) (B − z) −1 (3.19)
For z ∈ C and M ⊂ C we define
= (B − z) −1 (B − A) (A − z) −1 (3.20)
dist(z, M) = inf{|z − ζ|; ζ ∈ M} (3.21)
It is not hard to see that for any self adjoint operator A and any z ∈ ρ(A) the
operator norm ‖(A − z) −1 ‖ of the resolvent is given by
‖(A − z) −1 ‖ =
1
dist(z, σ(A)) . (3.22)
In particular, for a self adjoint operator A and z ∈ C \ R
‖(A − z) −1 ‖ ≤
1
Im z . (3.23)
For the rest of this section we assume that the operator A is bounded. In this case,
polynomials of the operator A can be defined straightforwardly
A 2 ϕ = A ( A(ϕ) ) (3.24)
(
A 3 ϕ = A A ( A(Aϕ) )) etc. (3.25)
More generally, if P is a complex valued polynomial in one real variable, P (λ) =
∑ n
j=0 a nλ j then
It is a key observation that
P (A) =
n∑
a n A j . (3.26)
j=0
‖P (A)‖ =
sup
λ∈σ(A)
|P (λ)| (3.27)
Let now f be a function in C ( σ(A) ) the complex-valued continuous functions on
(the compact set) σ(A). The Weierstraß approximation theorem tells us, that on
σ(A) the function f can be uniformly approximated by polynomials. Thus using
(3.27) we can define the operator f(A) as a norm limit of polynomials P n (A).
These operators satisfy