Linear Algebra Done Right · Problem 7A.31

Operators on Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · self-adjoint-operators

Problem

Fix a positive integer

n

. In the inner product space of continuous real-valued functions on

[-\pi, \pi]

with inner product

\langle f, g \rangle = \int_{-\pi}^{\pi} fg

, let

V = \operatorname{span}(1, \cos x, \cos 2x, \ldots, \cos nx, \sin x, \sin 2x, \ldots, \sin nx).

(a) Define

D \in \mathcal{L}(V)

Df = f'

. Show that

D^* = -D

. Conclude that

D

is normal but not self-adjoint. (b) Define

T \in \mathcal{L}(V)

Tf = f''

. Show that

T

is self-adjoint.