operator norm of multiplication operator on L2


The operator norm of the multiplication operator Mϕ is the essential supremumMathworldPlanetmath of the absolute valueMathworldPlanetmathPlanetmathPlanetmathPlanetmath of ϕ. (This may be expressed as Mϕop=ϕL.) In particular, if ϕ is essentially unboundedPlanetmathPlanetmath, 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 ψL2,

MϕψL2 = ψ(x)2ϕ(x)2𝑑μ(x)
(esssupϕ2)ψ(x)2𝑑μ(x)
= (esssup|ϕ|)ψL2

and, hence

Mϕop=supMϕψL2ψL2(esssup|ϕ|).

On the other hand, the operator norm bounds by the essential supremum of the absolute value . For any ϵ>0, the measureMathworldPlanetmathPlanetmath of the set

A={x|ϕ(x)|esssup|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 χB is the characteristic functionMathworldPlanetmathPlanetmathPlanetmath of B, we have

MϕχBL2 = ϕ(x)2χB(x)2𝑑μ(x)
= Bϕ(x)2𝑑μ(x)
μ(B)(esssup|ϕ|-ϵ)

and, hence

Mϕop=supMϕψL2ψL2MϕχBL2χBL2=esssup|ϕ|-ϵ.

Since this is true for every ϵ>0, we must have

Mϕopesssup|ϕ|.

Combining with the inequality in the opposite direction,

Mϕop=esssup|ϕ|.

It remains to consider the case where |ϕ| is essentially unbounded. This can be dealt with by a variation on the preceeding argumentPlanetmathPlanetmath.

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={xN+1|ϕ(x)|N}.

If μ(A)<, set B=A, otherwise let B be a subset of A whose measure is finite. Then, if χB is the characteristic function of B, we have

MϕχBL2 = ϕ(x)2χB(x)2𝑑μ(x)
= Bϕ(x)2𝑑μ(x)
μ(B)N

and, hence

Mϕop=supMϕψL2ψL2MϕχBL2χBL2=NR.

Since this is true for every R, we see that the operator norm is infiniteMathworldPlanetmathPlanetmath, i.e. the operator is unbounded.

Title operator norm of multiplication operator on L2
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