Notations. Let H be a Hilbert space. We denote by sp(T ) the spectrum of T and by sp

Université Denis Diderot 2012-2013

Mathématiques M1 Spectral Theory

Partiel

Durée 2 heures. Documents interdits.

Notations. Let H be a Hilbert space. We denote by sp(T ) the spectrum of T and by sp

(T ) the point spectrum of T .

1. Let a = (a

)

be a bounded sequence of complex numbers. Define T

e

= a

e

, where (e

)

is the canonical orthonormal basis of l

( N ).

(a) Show that T

extends uniquely to a linear bounded operator on l

(N).

(b) Compute Sp

(T

).

(c) Give a necessary and sufficient condition on a for T

to be invertible.

(d) Compute Sp(T

).

(e) Give a necessary and sufficient condition on a for T

to be compact.

2. Let q be a continuous function on [0, 1]

and µ a probability measure on [0, 1]. We define a bounded linear operator T

from C([0, 1]) to C([0, 1]) by

(T

f )(x) = Z

0

q(x, y)f (y)dµ(y) for all f ∈ C([0, 1]).

(a) Show that if q is a polynomial function, then T

is a finite rank operator on C([0, 1]).

(b) Deduce that T

is compact.

(c) Show that

kT

k = max

x∈[0,1]

Z

|q(x, y)|dµ(y)

3. Let H = L

([0, 1], λ) where λ is the Lebesgue measure. Define, for ξ ∈ H,

(T ξ)(x) = Z

0

y(1 − x)ξ(y)dy + Z

x

x(1 − y)ξ(y)dy.

(a) Show that T is a compact self-adjoint operator on H.

(b) Show that if ξ is continuous, then G = T ξ ∈ C

and G satisfies the differential equation G

= −ξ with the boundary conditions G(0) = G(1) = 0.

(c) Compute sp

(T ) and sp(T ).

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