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Spectral Theory of Operators
Spectral Theory of Operators
5 selected
Difficulty 6-8
5 unseen
View topic
Intermediate
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0 answered
2 intermediate
3 advanced
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Question 1 of 5
120s
intermediate (6/10)
compute
On
L
2
([
0
,
1
]
,
d
x
)
, the multiplication operator
(
M
g
f
)
(
x
)
=
g
(
x
)
f
(
x
)
for a bounded function
g
has what spectrum?
Hide and think first
A.
The interval
[
in
f
g
,
sup
g
]
on the real line, regardless of the measure-theoretic structure of
g
.
B.
The set of values of
g
on
[
0
,
1
]
, that is,
σ
(
M
g
)
=
g
([
0
,
1
])
, treated as a subset of
C
.
C.
A discrete set of eigenvalues
{
λ
n
}
n
=
1
∞
accumulating at zero, like a compact operator.
D.
The essential range of
g
,
σ
(
M
g
)
=
{
λ
:
Leb
(
g
−
1
(
B
(
λ
,
ϵ
)))
>
0
∀
ϵ
>
0
}
, as a subset of
C
.
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