operator norm of multiplication operator on L^2

operator norm of multiplication operator on

L

2

The operator norm of the

multiplication operator

M

ϕ

is the

essential supremum

of the

absolute value

of

ϕ

. (This may be
expressed as

∥

M

ϕ

∥

op

=

∥

ϕ

∥

L

∞

.)
In particular, if

ϕ

is essentially

unbounded

, the multiplication
operator is unbounded.

For the time being, assume that

ϕ

is

essentially bounded

.

On the one hand, the operator norm is

bounded

by the essential
supremum of the absolute value because, for any

ψ

∈

L

2

,

∥

M

ϕ

⁢

ψ

∥

L

2

=

∫

ψ

⁢

(

x

)

2

⁢

ϕ

⁢

(

x

)

2

⁢

𝑑

μ

⁢

(

x

)

≤

(

ess

⁢

sup

⁡

ϕ

2

)

⁢

∫

ψ

⁢

(

x

)

2

⁢

𝑑

μ

⁢

(

x

)

=

(

ess

⁢

sup

⁡

|

ϕ

|

)

⁢

∥

ψ

∥

L

2

and, hence

∥

M

ϕ

∥

op

=

sup

⁡

∥

M

ϕ

⁢

ψ

∥

L

2

∥

ψ

∥

L

2

≤

(

ess

⁢

sup

⁡

|

ϕ

|

)

.

On the other hand, the operator norm bounds by the essential supremum
of the absolute value . For any

ϵ

>

0

, the

measure

of the
set

A

=

{

x

∣

|

ϕ

⁢

(

x

)

|

≥

ess

⁢

sup

⁡

|

ϕ

|

-

ϵ

}

is

greater than zero

. If

μ

⁢

(

A

)

<

∞

, set

B

=

A

, otherwise let

B

be a subset of

A

whose measure is finite. Then, if

χ

B

is
the

characteristic function

of

B

, we have

∥

M

ϕ

⁢

χ

B

∥

L

2

=

∫

ϕ

⁢

(

x

)

2

⁢

χ

B

⁢

(

x

)

2

⁢

𝑑

μ

⁢

(

x

)

=

∫

B

ϕ

⁢

(

x

)

2

⁢

𝑑

μ

⁢

(

x

)

≥

μ

⁢

(

B

)

⁢

(

ess

⁢

sup

⁡

|

ϕ

|

-

ϵ

)

and, hence

∥

M

ϕ

∥

op

=

sup

⁡

∥

M

ϕ

⁢

ψ

∥

L

2

∥

ψ

∥

L

2

≥

∥

M

ϕ

⁢

χ

B

∥

L

2

∥

χ

B

∥

L

2

=

ess

⁢

sup

⁡

|

ϕ

|

-

ϵ

.

Since this is true for every

ϵ

>

0

, we must have

∥

M

ϕ

∥

op

≥

ess

⁢

sup

⁡

|

ϕ

|

.

Combining with the

inequality

in the opposite direction,

∥

M

ϕ

∥

op

=

ess

⁢

sup

⁡

|

ϕ

|

.

It remains to consider the case where

|

ϕ

|

is essentially
unbounded. This can be dealt with by a variation on the preceeding

argument

.

If

ϕ

is unbounded, then

μ

⁢

(

{

x

∣

|

ϕ

⁢

(

x

)

|

≥

R

}

)

>

0

for all

R

>

0

. Furthermore, for any

R

>

0

, we can find

N

>

R

such
that

μ

⁢

(

A

)

>

0

, where

A

=

{

x

∣

N

+

1

≥

|

ϕ

⁢

(

x

)

|

≥

N

}

.

If

μ

⁢

(

A

)

<

∞

, set

B

=

A

, otherwise let

B

be a subset of

A

whose measure is finite. Then, if

χ

B

is
the characteristic function of

B

, we have

∥

M

ϕ

⁢

χ

B

∥

L

2

=

∫

ϕ

⁢

(

x

)

2

⁢

χ

B

⁢

(

x

)

2

⁢

𝑑

μ

⁢

(

x

)

=

∫

B

ϕ

⁢

(

x

)

2

⁢

𝑑

μ

⁢

(

x

)

≥

μ

⁢

(

B

)

⁢

N

and, hence

∥

M

ϕ

∥

op

=

sup

⁡

∥

M

ϕ

⁢

ψ

∥

L

2

∥

ψ

∥

L

2

≥

∥

M

ϕ

⁢

χ

B

∥

L

2

∥

χ

B

∥

L

2

=

N

≥

R

.

Since this is true for every

R

, we see that the operator norm is

infinite

, i.e. the

operator

is unbounded.

Title

operator norm of multiplication operator on

L

2

Canonical name

OperatorNormOfMultiplicationOperatorOnL2

Date of creation

2013-04-06 22:14:23

Last modified on

2013-04-06 22:14:23

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

13

Author

rspuzio (6075)

Entry type

Theorem

Classification

msc 47B38

Generated on Fri Feb 9 21:47:13 2018 by

LaTeXML