MATH 520 Homework
Spring 2014
26. Define the translation operator T u(x) = u(x − 1) on L2 (R).
a) Find T ∗ .
b) Show that T is unitary.
c) Show that σ(T ) = σc (T ) = {λ ∈ C : |λ| = 1}.
Rx
27. Let T u(x) = 0 K(x, y)u(y) dy be a Volterra integral operator on L2 (0, 1) with a
bounded kernel, |K(x, y)| ≤ M . Show that σ(T ) = {0}. (There are several ways to
show that T has no nonzero eigenvalues. Here is one approach: Define the equivalent
norm on L2 (0, 1)
Z
1
u2 (x)e−2θx dx
||u||2θ =
0
and show that the supremum of
sufficiently large.)
||T u||θ
||u||θ
can be made arbitrarily small by choosing θ
28. Let T be the integral operator
Z
T u(x) =
1
(x + y)u(y) dy
0
on L2 (0, 1). Find σp (T ), σc (T ) and σr (T ) and the multiplicity of each eigenvalue.
29. Show that if S ∈ B(H) and T is compact, then T S and ST are also compact. (In
algebraic terms this means that the set of compact operators is an ideal in B(H).)
30. If T ∈ B(H) and T ∗ T is compact, show that T must be compact. Use this to show
that if T is compact then T ∗ must also be compact.