Analysis
Midterm 1
Write solutions in a neat and logical fashion, giving complete reasons for
all steps.
1. Let
H
be a complex Hilbert space.
(a) Prove that the set
F
(
H
) comprising all finiterank operators on
H
is a
minimal ideal in
B
(
H
).
(b) Prove that the set
K
(
H
) comprising all compact operators on
H
is a
minimal closed ideal in
B
(
H
).
Remark
: In both parts, the indefinite article may be replaced by a definite.
Solution
We shall not spell out the details that
F
(
H
) and
K
(
H
) are
ideals in
B
(
H
) as these are standard.
(a) Let
J
be a nonzero ideal in
B
(
H
). Choose a nonzero operator
A
∈
J
;
we may choose
ξ
∈
H
so that
η
:=
Aξ
is nonzero. Let
x, y
∈
H
be arbitrary
and choose
T
∈
B
(
H
) so that
Tη
=
y
; for example,
T
=
η

2
η
y
works.
Now compute: if
v
∈
H
then
TA
(
x
ξ
)(
v
) =
T
(
A x

v ξ
) =
x

v TAξ
=
x

v Tη
=
x

v y
= (
x
y
)(
v
)
which places
x
y
=
TA
(
x
ξ
) in the
ideal
J
. This shows that
J
contains
each rankone operator; finally, any finiterank operator is the sum of finitely
many rankone operators.
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 Fall '09
 Robinson
 Hilbert space, Compact operator, Eλ, complex Hilbert space

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