Measure, Integration & Real Analysis · Problem 11C.19

Probability Measures — Sheldon Axler

Mathematics — Measure Theory · convolution-operator-spectrum

Problem

Suppose

y > 0

. Define an operator

T

L^2(\mathbf{R})

Tf = f * P_y

. (a) Show that

T

is a self-adjoint operator on

L^2(\mathbf{R})

. (b) Show that

\mathrm{sp}(T) = [0, 1]

. [Because the spectrum of each compact operator is a countable set (by 10.93), part (b) above implies that

T

is not a compact operator. This conclusion differs from the situation on the unit circle—see Exercise 9 in Section 11B.]