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1. Let H be complex Hilbert space, A ∈ L(H) and P ∈ C [X ].

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Université Denis Diderot 2012-2013

Mathématiques M1 Spectral Theory

Examen du 16 janvier 2013 Durée 3 heures. Documents interdits.

1. Let H be complex Hilbert space, A ∈ L(H) and P ∈ C [X ].

(a) Recall the relation which exists between Sp(P(A)) and SpA for a self-adjoint operator A.

Show that the same result holds for an arbitrary bounded operator.

(b) Prove that if P (A) = 0, then SpA is contained in the set of roots of P .

2. Let H = L 2 ( R , λ) be the Hilbert space of complex-valued square integrable functions with respect to the Lebesgue measure. Let f : R → C be a measurable and λ-locally square integrable function on R (i.e. R

K |f | 2 dλ < ∞ for all K ⊂ R compact). Define the unbounded operator M on H : D(M ) = {ξ ∈ H :

Z

|f ξ | 2 dλ < ∞} (M ξ)(x) = f (x)ξ(x) ξ ∈ D(M ), x ∈ R .

(a) Show that M is densely defined.

(b) Let T be the unbounded operator on H defined by

D(T ) = D(M ) (T ξ)(x) = f (x)ξ(x) ξ ∈ D(T ), x ∈ R . Show that T is densely defined and M = T . Deduce that M is closed.

(c) Compute M .

(d) Show that sp(M ) = EssIm(f) where

EssIm(f ) = {λ ∈ C : ∀ > 0 λ(f −1 (B(λ, ))) > 0}.

(e) Show that if f is continuous then sp(M ) = f ( R ).

(f) Let A ⊂ R be a Borel subset such that λ(A) < ∞, ν be the finite measure on R defined by ν (B) = λ(A ∩ B) for all B Borel subset of R . Observe that the integral with respect to ν is

Z gdν =

Z

A

gdλ.

Let f (ν ) the measure image of ν by f i.e., f (ν )(B) = ν(f −1 (B)) for all Borel subset B ⊂ C . Let ξ = 1 A ∈ H . Suppose that f is bounded and continuous. Show that M is bounded and the spectral measure µ ξ of M associated to ξ is µ ξ = f (ν).

3. Let H be a complex Hilbert space and T ∈ L(H) be self-adjoint.

(a) Show that : ∀n ∈ N , ||T 2

n

|| = ||T || 2

n

.

(b) Show by recursion that the following property P(n) is true for any n ∈ N : P (n) : ∀k ∈ N such that 0 ≤ k ≤ 2 n , ||P k || = ||P || k . [Hint : for 2 n < k ≤ 2 n+1 , consider ` := 2 n+1 − k and P ` P k .]

Conclusion : we have shown that, for any self-adjoint operator P ∈ L(H), ||P n || = ||P|| n ,

∀n ∈ N .

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4. Let H be a complex Hilbert space. The goal of this exercise is to show that there are no bounded operators P, Q ∈ L(H) such that [P, Q] = 1 H (where 1 H ∈ L(H) is the identity operator) and [P, Q] := P Q − QP .

(a) Assume that H has a finite dimension n. Show, by using a simple argument, that there are no operators P, Q ∈ L(H) such that [P, Q] = 1 H .

(b) In the following we assume that the dimension of H is infinite. Show that, for any pair of operators P, Q ∈ L(H), we have :

∀n ∈ N , [P n , Q] =

n

X

j=1

P n−j [P, Q]P j−1 .

(c) We argue by contradiction and we assume that there exists operators P, Q ∈ L(H) such that [P, Q] = 1 H . By using the result of the previous question and of the previous exercise, show that, for any n ∈ N ,

2||P || ||Q|| ≥ n and conclude to a contradiction.

(d) Give an example of a complex Hilbert space H and two non bounded operators P and Q such that there exists a dense vector subspace V ⊂ H such that ∀ϕ ∈ V , [P, Q]ϕ = ϕ

5. Let H := ` 2 ( Z , C ) ' ` 2 ( Z ). We denote by ( n ) n∈ Z the canonical Hilbertian Hermitian orthogonal basis of H (i.e. n is the sequence which vanishes for all relative integer, excepted for n, for which it takes the value 1). We note L, R ∈ L(H) the operators defined by :

L n = n−1 et R n = n+1 , ∀n ∈ Z and A := L + R ∈ L(H).

(a) Compute L and R . Deduce that A is self-adjoint.

(b) We note U : ` 2 ( Z ) −→ L 2 ( R / Z ) the unitary operator defined by : (U n ) (θ) = e i2πnθ , ∀n ∈ Z (Fourier series isomorphism). For all function m ∈ L ( R / Z ) we note m b ∈ L L 2 ( R / Z )

the multiplication operator defined by :

∀f ∈ L 2 ( R / Z ), ( mf b ) (θ) = m(θ)f (θ), p.p.

Find U LU −1 and U RU −1 and show that they coincide with multiplication operators by functions to be precised. Deduce U AU −1 .

(c) Let ψ ∈ H be different of 0 and g := U ψ. We note F ψ := {P(A)ψ| P ∈ C [X]} = Vect C {A n ψ| n ∈ N }.

Show that the U F ψ , the image of F ψ by U , is equal to : P + g := {f g| f ∈ P + }, where P + is the subspace of L 2 ( R / Z ) of polynomials in e i2πθ and e −i2πθ which are even function of θ.

(d) Let ψ and g be as in the preceding question. We define h ∈ L 2 ( R / Z ) by : h(θ) = 2i sin(2πθ)g(−θ).

Show that, ∀f 1 , f 2 ∈ P + , hf 1 g, f 2 hi L

2

= 0.

(e) Deduce from the preceding questions that A does not admit a cyclic vector.

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