MAT 578 HW 9 Due Tuesday, 11/6/01
Choose three of the following problems. You may use the result of an unworked problem
to work a later problem. (
H
is a Hilbert space.)
1.
Let (
A
n
)
1
n
=1
be a monotone, bounded sequence of self adjoint operators in
B
(
H
). (I.e.,
either
A
1
A
2
or
A
1
A
2
, and for some constant
C>
0,
k
A
n
k
C
for all
n
.) Prove that there exists a self adjointoperator
A
2
B
(
H
)such that
A
n
!
A
in the
strong operator topology.(Hint: dene
A
by
h
Ax; x
i
and polarization.)
2.
Let
P
and
Q
be projections in
B
(
H
). Welet
P
^
Q
denote the projection onto
PH
\
QH
.
(i) Prove that if
P
and
Q
commute, then
P
^
Q
=
PQ
.
(ii) Prove that in general, (
PQ
)
n
!
P
^
Q
in the strong op erator topology.(Hint: show
that (
PQP
)
n
is a decreasing sequence of positive operators.)
3.
(i) Let
R
be a unital ring. Prove that for any
x
,
y
2
R
,
xy
+ 1 is invertible if and only
if
yx
+1 is invertible.
(ii) Let
a
,
b
be elements of a unital Banach algebra. Provethat
(
ab
)
[f
0
g
=
(
ba
)
[f
0
g
.
4.
(i) Let
A
and
B
be invertible positive operators in
B
(
H
), and suppose that
A
B
.
Prove that
A
1
=
2
B
1
=
2
. (Hints: First apply the fact
S
T
)
R
SR
R
TR
with
R
=
B
1
=
2
. Then use problem 3 to rearrange terms. Then apply the fact again.)
(ii) Prove that if 0
A
B
then
A
1
=
2
B
1
=
2
(without assuming invertibility). (Hint:
It's true for
A
+
and
B
+
.)
5.
Prove that it is NOT true that 0
A
B
implies
A
2
B
2
. (Hint: If it were, you
could apply it to
A
A
+
B
.)