Math 6210 - Homework 5
Due at 4 PM on 11/10/05
From Rudin: Chapter 5, # 1,2,3,4,5
All of the problems will examine linear operators on the Hilbert space H=`2(Z). Recall that
f∈Hif fis a function
f:Z−→ C
with X
n∈Z
|f(n)|2<∞and the inner product of f , g ∈His
(f, g) = X
n∈Z
f(n)g(n).
We also recall the definition of the adjoint T∗of a linear operator T. If Tis bounded then we saw
in class that for any g∈Hthe linear map
f7→ (T f, g )
is bounded. Therefore there is a unique element g0∈hsuch that (T f, g) = (f , g0) for all f∈H.
We define T∗g=g0.
If Tis unbounded it will in general only be defined on a subspace of Hwhich we denote dom T.
Then given any g∈Hthe map
f7→ (T f, g )
is only defined for f∈dom T. Furthermore it will not always be bounded. The subset of Hwhere
this map is bounded is the domain of the adjoint, dom T∗. Then for g∈dom T∗there is a unique
g0such that (T f, g) = (f , g0) for all f∈dom T. As in the bounded case we set T∗g=g0.
1. Define T:H−→ Hby (T f )(n) = f(n+ 1). Show that Tis bounded and that kTk= 1. Find
the adjoint T∗of T.
2. In this problem we will examine an unbounded operator S. Let dom Sbe the set of compactly
supported functions in H. That is f∈dom Sif f(n)6= 0 for only finitely many n∈Z. We
then set (Sf )(n) = nf (n).
(a) Show that Sis unbounded.
(b) Show that
dom S∗=(g∈Hsuch that X
n∈Z
n2|g(n)|2<∞).
Here are some hints. If the sum is <∞apply H¨olders inequality. If the sum is infinite
we need to show that g6∈ dom S∗by showing that
f7→ (Sf , g)
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