MAT 578 HW 6 Due Thursday, 10/11/01
Choose three of the following problems.
1.
Prove that if
f
is in
L
1
(
R
) and
g
is in
L
1
(
R
), then
f
g
is continuous. Use this to
prove that the Banach algebra
L
1
(
R
) does not have a unit.
2.
An element in a Banach algebra
A
is called a
commutator
if it can be written in the
form
ab
ba
for some
a
,
b
2
A
.Prove that if
A
is a unital Banach algebra, the unit is not
a commutator.
Outline: For
b
2
A
dene
D
b
2
B
(
A
)by
Dx
=
xb
bx
. Prove that
D
is a derivation:
D
(
xy
)=
x
(
Dy
)+(
Dx
)
y
.If
Da
= 1, compute
D
(
a
n
).
3.
Let
A
be a Banach algebra with unit, and let
D
be a closed subalgebra containing the
unit. We use
A
(
x
),
D
(
x
) for the sp ectrum of
x
in
A
,respectively
D
.Let
x
2
D
. Prove
the following:
(i)
A
(
x
)
D
(
x
).
(ii)
@
D
(
x
)
@
A
(
x
). (Hint: use problem 4 in assigment5.)
4.
Let
T
be the b ounded operator on three-dimensional complex Hilb ert space whose
matrix (relative to the standard basis) is
0
@
1 1 2
0 1 2
0 0 2
1
A
:
Let
f
(
z
)=
z
(2
z
) (and note that
f
(1) = 1,
f
(2) = 0).
(i) Compute
f
(
T
).
(ii) Compute
E
1
, the spectral pro jection of
T
for
f
1
g
.